copyright 1996 lawrence c. marsh powerpoint slides for undergraduate econometrics by lawrence c....
TRANSCRIPT
Copyright 1996 Lawrence C. Marsh
PowerPoint Slidesfor
Undergraduate Econometricsby
Lawrence C. Marsh
To accompany: Undergraduate Econometricsby R. Carter Hill, William E. Griffiths and George G. Judge
Publisher: John Wiley & Sons, 1997
Copyright 1996 Lawrence C. Marsh
The Role of Econometrics
in Economic Analysis
Chapter 1
Copyright © 1997 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein.
1.1
Copyright 1996 Lawrence C. Marsh
Using Information:
1. Information from economic theory.
2. Information from economic data.
The Role of Econometrics 1.2
Copyright 1996 Lawrence C. Marsh
Understanding Economic Relationships:
federalbudget
Dow-JonesStock Index
tradedeficit Federal Reserve
Discount Rate
capital gains taxrentcontrollaws
short termtreasury bills
power oflabor unions
crime rate
inflation
unemployment
money supply
1.3
Copyright 1996 Lawrence C. Marsh
economic theory
economic data } economicdecisions
To use information effectively:
*Econometrics* helps us combine economic theory and economic data .
Economic Decisions 1.4
Copyright 1996 Lawrence C. Marsh
Consumption, c, is some function of income, i :
c = f(i)
For applied econometric analysis this consumption function must be specified more precisely.
The Consumption Function 1.5
Copyright 1996 Lawrence C. Marsh
demand, qd, for an individual commodity:
qd = f( p, pc, ps, i )
supply, qs, of an individual commodity:
qs = f( p, pc, pf )
p = own price; pc = price of complements;ps = price of substitutes; i = income
p = own price; pc = price of competitive products;ps = price of substitutes; pf = price of factor inputs
demand
supply
1.6
Copyright 1996 Lawrence C. Marsh
Listing the variables in an economic relationship is not enough.
For effective policy we must know the amount of changeneeded for a policy instrument to bring about the desiredeffect:
How much ?How much ?
• By how much should the Federal Reserve raise interest rates to prevent inflation?
• By how much can the price of football tickets be increased and still fill the stadium?
1.7
Copyright 1996 Lawrence C. Marsh
Answering the How Much? question
Need to estimate parameters that are both:
1. unknown and
2. unobservable
1.8
Copyright 1996 Lawrence C. Marsh
Average or systematic behaviorover many individuals or many firms.
Not a single individual or single firm.
Economists are concerned with theunemployment rate and not whethera particular individual gets a job.
The Statistical Model 1.9
Copyright 1996 Lawrence C. Marsh
The Statistical Model
Actual vs. Predicted Consumption:
Actual = systematic part + random error
Systematic part provides prediction, f(i),but actual will miss by random error, e.
Consumption, c, is function, f, of income, i, with error, e:
c = f(i) + e
1.10
Copyright 1996 Lawrence C. Marsh
c = f(i) + e
Need to define f(i) in some way.
To make consumption, c, a linear function of income, i :
f(i) = 1 + 2 i
The statistical model then becomes:
c = 1 + 2 i + e
The Consumption Function 1.11
Copyright 1996 Lawrence C. Marsh
• Dependent variable, y, is focus of study (predict or explain changes in dependent variable).
• Explanatory variables, X2 and X3, help us explain
observed changes in the dependent variable.
y = 1 + 2 X2 + 3 X3 + e
The Econometric Model 1.12
Copyright 1996 Lawrence C. Marsh
Statistical Models
Controlled (experimental) vs.
Uncontrolled (observational)
Uncontrolled experiment (econometrics) explaining consump-
tion, y : price, X2, and income, X3, vary at the same time.
Controlled experiment (“pure” science) explaining mass, y :
pressure, X2, held constant when varying temperature, X3,
and vice versa.
1.13
Copyright 1996 Lawrence C. Marsh
Econometric model
• economic modeleconomic variables and parameters.
• statistical modelsampling process with its parameters.
• dataobserved values of the variables.
1.14
Copyright 1996 Lawrence C. Marsh
• Uncertainty regarding an outcome.• Relationships suggested by economic theory.• Assumptions and hypotheses to be specified.• Sampling process including functional form.• Obtaining data for the analysis.• Estimation rule with good statistical properties.• Fit and test model using software package.• Analyze and evaluate implications of the results.• Problems suggest approaches for further research.
The Practice of Econometrics 1.15
Copyright 1996 Lawrence C. Marsh
Note: the textbook uses the following symbol to mark sections with advanced material:
“Skippy”
1.16
Copyright 1996 Lawrence C. Marsh
Some Basic Probability Concepts
Chapter 2
Copyright © 1997 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein.
2.1
Copyright 1996 Lawrence C. Marsh
random variable: A variable whose value is unknown until it is observed.The value of a random variable results from an experiment.
The term random variable implies the existence of someknown or unknown probability distribution defined overthe set of all possible values of that variable.
In contrast, an arbitrary variable does not have aprobability distribution associated with its values.
Random Variable2.2
Copyright 1996 Lawrence C. Marsh
Controlled experiment values of explanatory variables are chosen with great care in accordance withan appropriate experimental design.
Uncontrolled experiment valuesof explanatory variables consist of nonexperimental observations overwhich the analyst has no control.
2.3
Copyright 1996 Lawrence C. Marsh
discrete random variable:A discrete random variable can take only a finitenumber of values, that can be counted by using the positive integers.
Example: Prize money from the followinglottery is a discrete random variable:
first prize: $1,000second prize: $50third prize: $5.75
since it has only four (a finite number) (count: 1,2,3,4) of possible outcomes:
$0.00; $5.75; $50.00; $1,000.00
Discrete Random Variable 2.4
Copyright 1996 Lawrence C. Marsh
continuous random variable:A continuous random variable can take any real value (not just whole numbers) in at least one interval on the real line.
Examples: Gross national product (GNP)money supplyinterest ratesprice of eggshousehold incomeexpenditure on clothing
Continuous Random Variable 2.5
Copyright 1996 Lawrence C. Marsh
A discrete random variable that is restrictedto two possible values (usually 0 and 1) iscalled a dummy variable (also, binary orindicator variable).
Dummy variables account for qualitative differences:gender (0=male, 1=female), race (0=white, 1=nonwhite),citizenship (0=U.S., 1=not U.S.), income class (0=poor, 1=rich).
Dummy Variable2.6
Copyright 1996 Lawrence C. Marsh
A list of all of the possible values takenby a discrete random variable along withtheir chances of occurring is called a probabilityfunction or probability density function (pdf).
die x f(x)one dot 1 1/6two dots 2 1/6three dots 3 1/6four dots 4 1/6five dots 5 1/6six dots 6 1/6
2.7
Copyright 1996 Lawrence C. Marsh
A discrete random variable X has pdf, f(x), which is the probabilitythat X takes on the value x.
f(x) = P(X=x)
0 < f(x) < 1
If X takes on the n values: x1, x2, . . . , xn, then f(x1) + f(x2)+. . .+f(xn) = 1.
Therefore,
2.8
Copyright 1996 Lawrence C. Marsh
Probability, f(x), for a discrete randomvariable, X, can be represented by height:
0 1 2 3 X
number, X, on Dean’s List of three roommates
f(x)0.2
0.4
0.1
0.3
2.9
Copyright 1996 Lawrence C. Marsh
A continuous random variable uses area under a curve rather than theheight, f(x), to represent probability:
f(x)
X$34,000 $55,000. .
per capita income, X, in the United States
0.13240.8676
red area
green area
2.10
Copyright 1996 Lawrence C. Marsh
Since a continuous random variable has an uncountably infinite number of values, the probability of one occurring is zero.
P [ X = a ] = P [ a < X < a ] = 0
Probability is represented by area.
Height alone has no area.
An interval for X is needed to get an area under the curve.
2.11
Copyright 1996 Lawrence C. Marsh
P [ a < X < b ] = f(x) dxb
a
The area under a curve is the integral ofthe equation that generates the curve:
For continuous random variables it is the integral of f(x), and not f(x) itself, whichdefines the area and, therefore, the probability.
2.12
Copyright 1996 Lawrence C. Marsh
n
Rule 2: axi = a xi i = 1 i = 1
n
Rule 1: xi = x1 + x2 + . . . + xni = 1
n
Rule 3: xi +yi = xi + yii = 1 i = 1 i = 1
n n n
Note that summation is a linear operatorwhich means it operates term by term.
Rules of Summation2.13
Copyright 1996 Lawrence C. Marsh
Rule 4: axi +byi = a xi + b yii = 1 i = 1 i = 1
n n n
Rules of Summation (continued)
Rule 5: x = xi =i = 1
n
n1 x1 + x2 + . . . + xn
n
The definition of x as given in Rule 5 impliesthe following important fact:
xi x) = 0i = 1
n
2.14
Copyright 1996 Lawrence C. Marsh
Rule 6: f(xi) = f(x1) + f(x2) + . . . + f(xn)i = 1
n
Notation: f(xi) = f(xi) = f(xi)
n
x i i = 1
n
Rule 7: f(xi,yj) = [ f(xi,y1) + f(xi,y2)+. . .+ f(xi,ym)] i = 1 i = 1
n m
j = 1
The order of summation does not matter :
f(xi,yj) = f(xi,yj)i = 1
n m
j = 1 j = 1
m n
i = 1
Rules of Summation (continued) 2.15
Copyright 1996 Lawrence C. Marsh
The mean or arithmetic average of arandom variable is its mathematicalexpectation or expected value, EX.
The Mean of a Random Variable
2.16
Copyright 1996 Lawrence C. Marsh
Expected Value
There are two entirely different, but mathematicallyequivalent, ways of determining the expected value:
1. Empirically: The expected value of a random variable, X,is the average value of the random variable in aninfinite number of repetitions of the experiment.
In other words, draw an infinite number of samples,and average the values of X that you get.
2.17
Copyright 1996 Lawrence C. Marsh
Expected Value
2. Analytically: The expected value of a discrete random variable, X, is determined by weighting all the possible values of X by the correspondingprobability density function values, f(x), and summing them up.
E[X] = x1f(x1) + x2f(x2) + . . . + xnf(xn)
In other words:
2.18
Copyright 1996 Lawrence C. Marsh
In the empirical case when the sample goes to infinity the values of X occur with a frequency equal to the corresponding f(x) in the analytical expression.
As sample size goes to infinity, the empirical and analytical methods will produce the same value.
Empirical vs. Analytical 2.19
Copyright 1996 Lawrence C. Marsh
x = xi
n
i = 1
where n is the number of sample observations.
Empirical (sample) mean:
E[X] = xif(xi)i = 1
n
where n is the number of possible values of xi.
Analytical mean:
Notice how the meaning of n changes.
2.20
Copyright 1996 Lawrence C. Marsh
E X = xi f(xi) i=1
n
The expected value of X-squared:
E X = xi f(xi) i=1
n2 2
It is important to notice that f(xi) does not change!
The expected value of X-cubed:
E X = xi f(xi) i=1
n3 3
The expected value of X:2.21
Copyright 1996 Lawrence C. Marsh
EX = 0 (.1) + 1 (.3) + 2 (.3) + 3 (.2) + 4 (.1)
2
EX = 0 (.1) + 1 (.3) + 2 (.3) + 3 (.2) + 4 (.1)22222
= 1.9
= 0 + .3 + 1.2 + 1.8 + 1.6
= 4.9
3
EX = 0 (.1) + 1 (.3) + 2 (.3) + 3 (.2) +4 (.1)33333
= 0 + .3 + 2.4 + 5.4 + 6.4
= 14.5
2.22
Copyright 1996 Lawrence C. Marsh
E [g(X)] = g(xi) f(xi)n
i = 1
g(X) = g1(X) + g2(X)
E [g(X)] = g1(xi) + g2(xi)] f(xi)n
i = 1
E [g(X)] = g1(xi) f(xi) + g2(xi) f(xi)n
i = 1
n
i = 1
E [g(X)] = E [g1(X)] + E [g2(X)]
2.23
Copyright 1996 Lawrence C. Marsh
Adding and Subtracting Random Variables
E(X-Y) = E(X) - E(Y)
E(X+Y) = E(X) + E(Y)
2.24
Copyright 1996 Lawrence C. Marsh
E(X+a) = E(X) + a
Adding a constant to a variable willadd a constant to its expected value:
Multiplying by constant will multiply its expected value by that constant:
E(bX) = b E(X)
2.25
Copyright 1996 Lawrence C. Marsh
var(X) = average squared deviations around the mean of X.
var(X) = expected value of the squared deviations around the expected value of X.
var(X) = E [(X - EX) ] 2
Variance2.26
Copyright 1996 Lawrence C. Marsh
var(X) = E [(X - EX) ]
= E [X - 2XEX + (EX) ]
2
2
2= E(X ) - 2 EX EX + E (EX)
2
2
= E(X ) - 2 (EX) + (EX) 2 2 2
= E(X ) - (EX) 2 2
var(X) = E [(X - EX) ] 2
var(X) = E(X ) - (EX) 22
2.27
Copyright 1996 Lawrence C. Marsh
variance of a discreterandom variable, X:
standard deviation is square root of variance
var ( X ) = (xi - EX )2 f(xi)i = 1
n
ه
2.28
Copyright 1996 Lawrence C. Marsh
xi f(xi) (xi - EX) (xi - EX) f(xi)
2 .1 2 - 4.3 = -2.3 5.29 (.1) = .5293 .3 3 - 4.3 = -1.3 1.69 (.3) = .5074 .1 4 - 4.3 = - .3 .09 (.1) = .0095 .2 5 - 4.3 = .7 .49 (.2) = .0986 .3 6 - 4.3 = 1.7 2.89 (.3) = .867
xi f(xi) = .2 + .9 + .4 + 1.0 + 1.8 = 4.3
(xi - EX) f(xi) = .529 + .507 + .009 + .098 + .867 = 2.01
2
2
calculate the variance for a discrete random variable, X:
i = 1
n
n
i = 1
2.29
Copyright 1996 Lawrence C. Marsh
Z = a + cX
var(Z) = var(a + cX)
= E [(a+cX) - E(a+cX)]
= c var(X)
2
2
var(a + cX) = c var(X)2
2.30
Copyright 1996 Lawrence C. Marsh
A joint probability density function, f(x,y), provides the probabilities associated with the joint occurrence of all of the possible pairs of X and Y.
Joint pdf 2.31
Copyright 1996 Lawrence C. Marsh
college gradsin household
.15
.05
.45
.35
joint pdff(x,y)
Y = 1 Y = 2
vacationhomesowned
X = 0
X = 1
Survey of College City, NY
f(0,1) f(0,2)
f(1,1) f(1,2)
2.32
Copyright 1996 Lawrence C. Marsh
E[g(X,Y)] = g(xi,yj) f(xi,yj)i j
E(XY) = (0)(1)(.45)+(0)(2)(.15)+(1)(1)(.05)+(1)(2)(.35)=.75
E(XY) = xi yj f(xi,yj)i j
Calculating the expected value of functions of two random variables.
2.33
Copyright 1996 Lawrence C. Marsh
The marginal probability density functions,f(x) and f(y), for discrete random variables,can be obtained by summing over the f(x,y) with respect to the values of Y to obtain f(x) with respect to the values of X to obtain f(y).
f(xi) = f(xi,yj) f(yj) = f(xi,yj)ij
Marginal pdf 2.34
Copyright 1996 Lawrence C. Marsh
.15
.05
.45
.35
marginalY = 1 Y = 2
X = 0
X = 1
.60
.40
.50.50
f(X = 1)
f(X = 0)
f(Y = 1) f(Y = 2)
marginalpdf for Y:
marginalpdf for X:
2.35
Copyright 1996 Lawrence C. Marsh
The conditional probability density
functions of X given Y=y , f(x|y),
and of Y given X=x , f(y|x), are obtained by dividing f(x,y) by f(y)
to get f(x|y) and by f(x) to get f(y|x).
f(x|y) = f(y|x) =f(x,y) f(x,y)f(y) f(x)
Conditional pdf 2.36
Copyright 1996 Lawrence C. Marsh
.15
.05
.45
.35
conditonalY = 1 Y = 2
X = 0
X = 1
.60
.40
.50.50
.25.75
.875.125
.90.10 .70
.30
f(Y=2|X= 0)=.25f(Y=1|X = 0)=.75
f(Y=2|X = 1)=.875
f(X=0|Y=2)=.30
f(X=1|Y=2)=.70
f(X=0|Y=1)=.90
f(X=1|Y=1)=.10
f(Y=1|X = 1)=.125
2.37
Copyright 1996 Lawrence C. Marsh
X and Y are independent random variables if their joint pdf, f(x,y),is the product of their respectivemarginal pdfs, f(x) and f(y) .
f(xi,yj) = f(xi) f(yj)for independence this must hold for all pairs of i and j
Independence 2.38
Copyright 1996 Lawrence C. Marsh
.15
.05
.45
.35
not independentY = 1 Y = 2
X = 0
X = 1
.60
.40
.50.50
f(X = 1)
f(X = 0)
f(Y = 1) f(Y = 2)
marginalpdf for Y:
marginalpdf for X:
.50x.60=.30 .50x.60=.30
.50x.40=.20 .50x.40=.20 The calculations in the boxes show the numbers required to have independence.
2.39
Copyright 1996 Lawrence C. Marsh
The covariance between two randomvariables, X and Y, measures thelinear association between them.
cov(X,Y) = E[(X - EX)(Y-EY)]
Note that variance is a special case of covariance.
cov(X,X) = var(X) = E[(X - EX) ]2
Covariance2.40
Copyright 1996 Lawrence C. Marsh
cov(X,Y) = E [(X - EX)(Y-EY)]
= E [XY - X EY - Y EX + EX EY]
= E(XY) - 2 EX EY + EX EY
= E(XY) - EX EY
cov(X,Y) = E [(X - EX)(Y-EY)]
cov(X,Y) = E(XY) - EX EY
= E(XY) - EX EY - EY EX + EX EY
2.41
Copyright 1996 Lawrence C. Marsh
.15
.05
.45
.35
Y = 1 Y = 2
X = 0
X = 1
.60
.40
.50.50
EX=0(.60)+1(.40)=.40
EY=1(.50)+2(.50)=1.50
E(XY) = (0)(1)(.45)+(0)(2)(.15)+(1)(1)(.05)+(1)(2)(.35)=.75
EX EY = (.40)(1.50) = .60
cov(X,Y) = E(XY) - EX EY = .75 - (.40)(1.50) = .75 - .60 = .15
covariance
2.42
Copyright 1996 Lawrence C. Marsh
The correlation between two random variables X and Y is their covariance divided by the square roots of their respective variances.
Correlation is a pure number falling between -1 and 1.
cov(X,Y)(X,Y) =var(X) var(Y)
Correlation2.43
Copyright 1996 Lawrence C. Marsh
.15
.05
.45
.35
Y = 1 Y = 2
X = 0
X = 1
.60
.40
.50.50
EX=.40
EY=1.50
cov(X,Y) = .15
correlation
EX=0(.60)+1(.40)=.4022 2
var(X) = E(X ) - (EX) = .40 - (.40) = .24
2 2
2
EY=1(.50)+2(.50) = .50 + 2.0 = 2.50
2 2 2
var(Y) = E(Y ) - (EY) = 2.50 - (1.50) = .25
2 2
2
(X,Y) =cov(X,Y)
var(X) var(Y)
(X,Y) = .61
2.44
Copyright 1996 Lawrence C. Marsh
Independent random variables have zero covariance and, therefore, zero correlation.
The converse is not true.
Zero Covariance & Correlation2.45
Copyright 1996 Lawrence C. Marsh
The expected value of the weighted sumof random variables is the sum of the expectations of the individual terms.
Since expectation is a linear operator,it can be applied term by term.
E[c1X + c2Y] = c1EX + c2EY
E[c1X1+...+ cnXn] = c1EX1+...+ cnEXn
In general, for random variables X1, . . . , Xn :
2.46
Copyright 1996 Lawrence C. Marsh
The variance of a weighted sum of random variables is the sum of the variances, each times the square of the weight, plus twice the covariances of all the random variables times the products of their weights.
var(c1X + c2Y)=c1 var(X)+c2 var(Y) + 2c1c2cov(X,Y)2 2
var(c1X c2Y) = c1 var(X)+c2 var(Y) 2c1c2cov(X,Y)2 2
Weighted sum of random variables:
Weighted difference of random variables:
2.47
Copyright 1996 Lawrence C. Marsh
The Normal Distribution
Y ~ N(,2)
f(y) =2 2
1 exp
y
f(y)
2 2
(y - )2-
2.48
Copyright 1996 Lawrence C. Marsh
The Standardized Normal
Z ~ N(,)
f(z) =2
1 exp 2z2-
Z = (y - )/
2.49
Copyright 1996 Lawrence C. Marsh
P [ Y > a ] = P > = P Z > a - a - Y -
y
f(y)
a
Y ~ N(,2)2.50
Copyright 1996 Lawrence C. Marsh
P [ a < Y < b ] = P < <
= P < Z <
a - Y -
b -
a -
b -
y
f(y)
a
Y ~ N(,2)
b
2.51
Copyright 1996 Lawrence C. Marsh
Y1 ~ N(1,12), Y2 ~ N(2,2
2), . . . , Yn ~ N(n,n2)
W = c1Y1 + c2Y2 + . . . + cnYn
Linear combinations of jointlynormally distributed random variablesare themselves normally distributed.
W ~ N[ E(W), var(W) ]
2.52
Copyright 1996 Lawrence C. Marsh
mean: E[V] = E[ (m) ] = m
If Z1, Z2, . . . , Zm denote m independentN(0,1) random variables, andV = Z1 + Z2 + . . . + Zm , then V ~ (m)
2 2 2 2
V is chi-square with m degrees of freedom.
Chi-Square
variance: var[V] = var[ (m) ] = 2m
If Z1, Z2, . . . , Zm denote m independentN(0,1) random variables, andV = Z1 + Z2 + . . . + Zm , then V ~ (m)
2 2 2 2
V is chi-square with m degrees of freedom.
2
2
2.53
Copyright 1996 Lawrence C. Marsh
mean: E[t] = E[t(m) ] = 0 symmetric about zero
variance: var[t] = var[t(m) ] = m / (m2)
If Z ~ N(0,1) and V ~ (m) and if Z and Vare independent then, ~ t(m)
t is student-t with m degrees of freedom.
2
t = Z
V m
Student - t 2.54
Copyright 1996 Lawrence C. Marsh
If V1 ~ (m1) and V2 ~ (m
2) and if V1 and V2
are independent, then~ F(m
1,m
2)
F is an F statistic with m1 numeratordegrees of freedom and m2 denominatordegrees of freedom.
2
F =V1
m1
V2m
2
2
F Statistic 2.55
Copyright 1996 Lawrence C. Marsh
The Simple Linear Regression
Model
Chapter 3
Copyright © 1997 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein.
3.1
Copyright 1996 Lawrence C. Marsh
1. Estimate a relationship among economic variables, such as y = f(x).
2. Forecast or predict the value of one variable, y, based on the value of another variable, x.
Purpose of Regression Analysis
3.2
Copyright 1996 Lawrence C. Marsh
Weekly Food Expenditures
y = dollars spent each week on food items.
x = consumer’s weekly income.
The relationship between x and the expected value of y , given x, might be linear:
E(y|x) = 1 + 2 x
3.3
Copyright 1996 Lawrence C. Marsh
f(y|x=480)
f(y|x=480)
yy|x=480
Figure 3.1a Probability Distribution f(y|x=480) of Food Expenditures if given income x=$480.
3.4
Copyright 1996 Lawrence C. Marshf(y|x) f(y|x=480) f(y|x=800)
yy|x=480 y|x=800
Figure 3.1b Probability Distribution of FoodExpenditures if given income x=$480 and x=$800.
3.5
Copyright 1996 Lawrence C. Marsh
{1
x
E(y|x)
E(y|x)
AverageExpenditure
x (income)
E(y|x)=1+2x
2=E(y|x)
x
Figure 3.2 The Economic Model: a linear relationship between avearage expenditure on food and income.
3.6
Copyright 1996 Lawrence C. Marsh
..
xtx1=480 x2=800
y tf(yt)
Figure 3.3. The probability density function for yt at two levels of household income, x t
expe
nditu
re
Homoskedastic Case
income
3.7
Copyright 1996 Lawrence C. Marsh
.
x tx1 x2
y tf(yt)
Figure 3.3+. The variance of yt increases as household income, x t , increases.
expe
nditu
re
Heteroskedastic Case
x3
..
income
3.8
Copyright 1996 Lawrence C. Marsh
Assumptions of the Simple Linear Regression Model - I
1. The average value of y, given x, is given bythe linear regression: E(y) = 1 + 2x
2. For each value of x, the values of y aredistributed around their mean with variance: var(y) = 2
3. The values of y are uncorrelated, having zerocovariance and thus no linear relationship: cov(yi ,yj) = 0
4. The variable x must take at least two differentvalues, so that x ° c, where c is a constant.
3.9
Copyright 1996 Lawrence C. Marsh
5. (optional) The values of y are normally distributed about their mean for each value of x:
y ~ N [(1+2x),2 ]
One more assumption that is often used inpractice but is not required for least squares:
3.10
Copyright 1996 Lawrence C. Marsh
The Error Term
y is a random variable composed of two parts:
I. Systematic component: E(y) = 1 + 2x This is the mean of y.
II. Random component: e = y - E(y) = y - 1 - 2x This is called the random error.
Together E(y) and e form the model: y = 1 + 2x + e
3.11
Copyright 1996 Lawrence C. Marsh
Figure 3.5 The relationship among y, e and the true regression line.
.
..
.
y4
y1
y2
y3
x1 x2 x3 x4
}
}
{
{
e1
e2
e3
e4 E(y) = 1 + 2x
x
y 3.12
Copyright 1996 Lawrence C. Marsh
}.
}.
.
.
y4
y1
y2 y3
x1 x2 x3 x4
{
{
e1
e2
e3
e4
x
y
Figure 3.7a The relationship among y, e and the fitted regression line.
^
y = b1 + b2x^
.
.
.
.y1
y2
y3
y4
^^
^^
^^
^^
3.13
Copyright 1996 Lawrence C. Marsh
{
{.
.
.
.
.
y4
y1
y2 y3
x1 x2 x3 x4 x
y
Figure 3.7b The sum of squared residuals from any other line will be larger.
y = b1 + b2x^
.
..y1
^y3^
y4^ y = b1 + b2x
^* * **
e1^*
e2^*
y2^*
e3^*
*e4*
*{{
3.14
Copyright 1996 Lawrence C. Marshf(.) f(e) f(y)
Figure 3.4 Probability density function for e and y
0 1+2x
3.15
Copyright 1996 Lawrence C. MarshThe Error Term Assumptions
1. The value of y, for each value of x, is y = 1 + 2x + e
2. The average value of the random error e is: E(e) = 0
3. The variance of the random error e is: var(e) = 2 = var(y)
4. The covariance between any pair of e’s is: cov(ei ,ej) = cov(yi ,yj) = 0
5. x must take at least two different values so that x ° c, where c is a constant.
6. e is normally distributed with mean 0, var(e)=2 (optional) e ~ N(0,2)
3.16
Copyright 1996 Lawrence C. MarshUnobservable Nature
of the Error Term
1. Unspecified factors / explanatory variables, not in the model, may be in the error term.
2. Approximation error is in the error term if relationship between y and x is not exactly a perfectly linear relationship.
3. Strictly unpredictable random behavior that may be unique to that observation is in error.
3.17
Copyright 1996 Lawrence C. Marsh
Population regression values: y t = 1 + 2x t + e t
Population regression line: E(y t|x t) = 1 + 2x t
Sample regression values: y t = b1 + b2x t + e t
Sample regression line: y t = b1 + b2x t
^
^
3.18
Copyright 1996 Lawrence C. Marsh
y t = 1 + 2x t + e t
Minimize error sum of squared deviations:
S(1,2) = (y t - 1 - 2x t )2 (3.3.4)t=1
T
e t = y t - 1 - 2x t
3.19
Copyright 1996 Lawrence C. MarshMinimize w. r. t. 1 and 2:
S(1,2) = (y t - 1 - 2x t )2 (3.3.4)t =1
T
= - 2 (y t - 1 - 2x t )
= - 2 x t (y t - 1 - 2x t )
S()1
S()2
Set each of these two derivatives equal to zero and
solve these two equations for the two unknowns: 1 2
3.20
Copyright 1996 Lawrence C. Marsh
S(.)S(.)
ibi
..
.
Minimize w. r. t. 1 and 2:
S() = (y t - 1 - 2x t )2t =1
T
S(.)i
< 0 S(.)
i
> 0 S(.)i
= 0
3.21
Copyright 1996 Lawrence C. Marsh To minimize S(.), you set the two
derivatives equal to zero to get:
= - 2 (y t - b1 - b2x t ) = 0
= - 2 x t (y t - b1 - b2x t ) = 0
S()1
S()2
When these two terms are set to zero,
1 and 2 become b1 and b2 because they no longer
represent just any value of 1 and 2 but the special
values that correspond to the minimum of S() .
3.22
Copyright 1996 Lawrence C. Marsh
- 2 (y t - b1 - b2x t ) = 0
- 2 x t (y t - b1 - b2x t ) = 0
y t - Tb1 - b2 x t = 0
x t y t - b1 x t - b2 xt = 0 2
Tb1 + b2 x t = y t
b1 x t + b2 xt = x t y t 2
3.23
Copyright 1996 Lawrence C. Marsh
Solve for b1 and b2 using definitions of x and y
Tb1 + b2 x t = y t
b1 x t + b2 xt = x t y t 2
T x t yt - x t y t
T x t - ( x t)2 2
b2 =
b1 = y - b2 x
3.24
Copyright 1996 Lawrence C. Marsh
elasticities
percentage change in ypercentage change in x
= = x/xy/y
= y xx y
Using calculus, we can get the elasticity at a point:
= lim = y xx y
y xx yx 0
3.25
Copyright 1996 Lawrence C. Marsh
E(y) = 1 + 2 x
E(y)x = 2
applying elasticities
E(y)x
= 2=
E(y)x
E(y)x
3.26
Copyright 1996 Lawrence C. Marsh
estimating elasticities yx = b2
= yx
yx^
yt = b1 + b2 x t = 4 + 1.5 x t^
x = 8 = average number of years of experiencey = $10 = average wage rate
= 1.5 = 1.2810= b2
yx^
3.27
Copyright 1996 Lawrence C. Marsh
Prediction
yt = 4 + 1.5 x t^
Estimated regression equation:
x t = years of experience
yt = predicted wage rate^
If x t = 2 years, then yt = $7.00 per hour.^
If x t = 3 years, then yt = $8.50 per hour.^
3.28
Copyright 1996 Lawrence C. Marsh
log-log models
ln(y) = 1 + 2 ln(x)
ln(y)x
ln(x)x= 2
yx = 2
1y
xx
1x
3.29
Copyright 1996 Lawrence C. Marsh
yx = 2
1y
xx
1x
= 2
yx
xy
elasticity of y with respect to x:
= 2
yx
xy
=
3.30
Copyright 1996 Lawrence C. Marsh
Properties of Least Squares
Estimators
Chapter 4
Copyright © 1997 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein.
4.1
Copyright 1996 Lawrence C. Marsh
yt = household weekly food expenditures
Simple Linear Regression Model
yt = 1 + 2 x t + t
x t = household weekly income
For a given level of x t, the expected
level of food expenditures will be:
E(yt|x t) = 1 + 2 x t
4.2
Copyright 1996 Lawrence C. Marsh
1. yt = 1 + 2x t + t
2. E( t) = 0 <=> E(yt) = 1 + 2x t
3. var( t) = 2 = var(yt)
4. cov( i, j) = cov(yi,yj) = 0
5. x t ° c for every observation
6. t~N(0,2) <=> yt~N(1+ 2x t,
2)
Assumptions of the SimpleLinear Regression Model
4.3
Copyright 1996 Lawrence C. MarshThe population parameters 1 and 2
are unknown population constants.
The formulas that produce thesample estimates b1 and b2 are
called the estimators of 1 and 2.
When b0 and b1 are used to represent
the formulas rather than specific values,
they are called estimators of 1 and 2which are random variables becausethey are different from sample to sample.
4.4
Copyright 1996 Lawrence C. Marsh
• If the least squares estimators b0 and b1
are random variables, then what are theirtheir means, variances, covariances andprobability distributions?
• Compare the properties of alternative estimators to the properties of the least squares estimators.
Estimators are Random Variables ( estimates are not )
4.5
Copyright 1996 Lawrence C. Marsh
The Expected Values of b1 and b2
The least squares formulas (estimators)in the simple regression case:
b2 =Txtyt - xt yt
Txt -(xt)2 2
b1 = y - b2x
where y = yt / T and x = x t / T
(3.3.8a)
(3.3.8b)
4.6
Copyright 1996 Lawrence C. MarshSubstitute in yt = 1 + 2x t + t
to get:
b2 = 2 +Txtt - xt t
Txt -(xt)2 2
The mean of b2 is:
Eb2 = 2 +TxtEt - xt Et
Txt -(xt)2 2
Since Et = 0, then Eb2 = 2 .
4.7
Copyright 1996 Lawrence C. Marsh
The result Eb2 = 2 means that
the distribution of b2 is centered at 2.
Since the distribution of b2
is centered at 2 ,we say that
b2 is an unbiased estimator of 2.
An Unbiased Estimator 4.8
Copyright 1996 Lawrence C. Marsh
The unbiasedness result on the previous slide assumes that weare using the correct model.
If the model is of the wrong formor is missing important variables,
then Et ° 0, then Eb2 ° 2 .
Wrong Model Specification 4.9
Copyright 1996 Lawrence C. Marsh
Unbiased Estimator of the Intercept
In a similar manner, the estimator b1
of the intercept or constant term can beshown to be an unbiased estimator of 1
when the model is correctly specified.
Eb1 = 1
4.10
Copyright 1996 Lawrence C. Marsh
b2 =Txtyt xt yt
Txt (xt)2 2 (3.3.8a)
(4.2.6)
Equivalent expressions for b2:
Expand and multiply top and bottom by T:
b2 =(xt x )yt y )xt x )
2
4.11
Copyright 1996 Lawrence C. Marsh
Variance of b2
Given that both yt and t have variance 2,
the variance of the estimator b2 is:
b2 is a function of the yt values butvar(b2) does not involve yt directly.
x t x
2
2var(b2) =
4.12
Copyright 1996 Lawrence C. Marsh
Variance of b1
x t x2var(b1) = 2
x t2
the variance of the estimator b1 is:
b1 = y b2xGiven
4.13
Copyright 1996 Lawrence C. Marsh
Covariance of b1 and b2
x t x2cov(b1,b2) = 2
x
If x = 0, slope can change without affectingthe variance.
4.14
Copyright 1996 Lawrence C. Marsh What factors determine variance and covariance ?
1. 2: uncertainty about yt values uncertainty about b1, b2 and their relationship.
2. The more spread out the xt values are then the more confidence we have in b1, b2, etc.
3. The larger the sample size, T, the smaller the variances and covariances.
4. The variance b1 is large when the (squared) xt values are far from zero (in either direction).
5. Changing the slope, b2, has no effect on the intercept, b1, when the sample mean is zero. But if sample mean is positive, the covariance between b1 and b2 will be negative, and vice versa.
4.15
Copyright 1996 Lawrence C. Marsh
Gauss-Markov Theorm
Under the first five assumptions of the simple, linear regression model, the ordinary least squares estimators b1 and b2 have the smallest variance of all linear and unbiased estimators of1 and 2. This means that b1and b2 are the Best Linear Unbiased Estimators (BLUE) of 1 and 2.
4.16
Copyright 1996 Lawrence C. Marsh
implications of Gauss-Markov
1. b1 and b2 are “best” within the class of linear and unbiased estimators.
2. “Best” means smallest variance within the class of linear/unbiased.
3. All of the first five assumptions must hold to satisfy Gauss-Markov.
4. Gauss-Markov does not require assumption six: normality.
5. G-Markov is not based on the least squares principle but on b1 and b2.
4.17
Copyright 1996 Lawrence C. Marsh
G-Markov implications (continued)
6. If we are not satisfied with restricting our estimation to the class of linear and unbiased estimators, we should ignore the Gauss-Markov Theorem and use some nonlinear and/or biased estimator instead. (Note: a biased or nonlinear estimator could have smaller variance than those satisfying Gauss-Markov.)
7. Gauss-Markov applies to the b1 and b2
estimators and not to particular sample values (estimates) of b1 and b2.
4.18
Copyright 1996 Lawrence C. MarshProbability Distribution
of Least Squares Estimators
b2 ~ N 2 , x t x
2
2
b1 ~ N 1 ,x t x2
2 x t
2
4.19
Copyright 1996 Lawrence C. Marsh yt and t normally
distributed The least squares estimator of 2 can beexpressed as a linear combination of yt’s:
b2 = wt yt
b1 = y b2xx t x
2where wt =x t x
This means that b1and b2 are normal sincelinear combinations of normals are normal.
4.20
Copyright 1996 Lawrence C. Marsh
normally distributed under The Central Limit Theorem
If the first five Gauss-Markov assumptionshold, and sample size, T, is sufficiently large,then the least squares estimators, b1 and b2,have a distribution that approximates thenormal distribution with greater accuracythe larger the value of sample size, T.
4.21
Copyright 1996 Lawrence C. Marsh
Consistency
We would like our estimators, b1 and b2, to collapse onto the true population values, 1 and 2, as sample size, T, goes to infinity.
One way to achieve this consistency property is for the variances of b1 and b2 to go to zero as T goes to infinity.
Since the formulas for the variances of the least squares estimators b1 and b2 show that their variances do, in fact, go to zero, then b1 and b2, are consistent estimators of 1 and 2.
4.22
Copyright 1996 Lawrence C. Marsh Estimating the variance
of the error term, 2
et = yt b1 b2 x t^
et
^t =1
T 2
T2
=
is an unbiased estimator of
2
^
^
4.23
Copyright 1996 Lawrence C. Marsh
The Least Squares Predictor, yo ^
Given a value of the explanatory variable, Xo, we would like to predicta value of the dependent variable, yo.
The least squares predictor is:
yo = b1 + b2 x o (4.7.2)
^
4.24
Copyright 1996 Lawrence C. Marsh
Inference in the Simple
Regression Model
Chapter 5
Copyright © 1997 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein.
5.1
Copyright 1996 Lawrence C. Marsh
1. yt = 1 + 2x t + t
2. E( t) = 0 <=> E(yt) = 1 + 2x t
3. var( t) = 2 = var(yt)
4. cov( i, j) = cov(yi,yj) = 0
5. x t ° c for every observation
6. t~N(0,2) <=> yt~N(1+ 2x t,
2)
Assumptions of the Simple Linear Regression Model
5.2
Copyright 1996 Lawrence C. Marsh
Probability Distribution of Least Squares
Estimators
b1 ~ N 1 ,x t x2
2 x t2
b2 ~ N 2 , x t x2
2
5.3
Copyright 1996 Lawrence C. Marsh
2^ =
et^
Unbiased estimator of the error variance:
2
2
^
Transform to a chi-square distribution:
Error Variance Estimation 5.4
Copyright 1996 Lawrence C. MarshWe make a correct decision if:
• The null hypothesis is false and we decide to reject it.
• The null hypothesis is true and we decide not to reject it.
Our decision is incorrect if:
• The null hypothesis is true and we decide to reject it. This is a type I error.
• The null hypothesis is false and we decide not to reject it. This is a type II error.
5.5
Copyright 1996 Lawrence C. Marsh
b2 ~ N 2 , x t x2
2
Create a standardized normal random variable, Z, by subtracting the mean of b2 and dividing by its standard deviation:
b2 2
var(b2)
5.6
Copyright 1996 Lawrence C. Marsh
Simple Linear Regression
yt = 1 + 2x t + twhere E t = 0
yt ~ N(1+ 2x t , 2)
since Eyt = 1 + 2x t
t = yt 1 2x t
Therefore, t ~ N(0,2) .
5.7
Copyright 1996 Lawrence C. Marsh
Create a Chi-Square
t ~ N(0,2) but want N(0,) .
t /~ N(0,) Standard Normal .
t /~ Chi-Square .
5.8
Copyright 1996 Lawrence C. Marsh
Sum of Chi-Squares
t =1 t /=
1 / 2 / T
/
+
+. . .+ =
Therefore, t =1 t /
5.9
Copyright 1996 Lawrence C. Marsh
Since the errors t = yt 1 2x t
are not observable, we estimate them with
the sample residuals e t = yt b1 b2x t.
Unlike the errors, the sample residuals arenot independent since they use up two degreesof freedom by using b1 and b2 to estimate 1 and 2.
We get only T2 degrees of freedom instead of T.
Chi-Square degrees of freedom5.10
Copyright 1996 Lawrence C. Marsh
Student-t Distribution
t = ~ t(m)
Z
V / m
where Z ~ N(0,1)
and V ~ (m)
2
5.11
Copyright 1996 Lawrence C. Marsh
t = ~ t(m)
Z
V / ( T2)
where Z = (b2 2)
var(b2)
and var(b2) =
2
( xi x )2
5.12
Copyright 1996 Lawrence C. Marsh
t = Z
V / (T-2)
(b2 2)
var(b2)t =
(T2) 2
2
^( T2)
V = (T2)
2
2
^
5.13
Copyright 1996 Lawrence C. Marsh
var(b2) =
2
( xi x )2
(b2 2)
2
( xi x )2
t = =
(T2) 2
2
^( T2)
(b2 2)
2
( xi x )2
^
notice thecancellations
5.14
Copyright 1996 Lawrence C. Marsh
(b2 2)
2
( xi x )2
^t = =
(b2 2)
var(b2)^
t = (b2 2)
se(b2)
5.15
Copyright 1996 Lawrence C. Marsh
Student’s t - statistic
t = ~ t (T2)
(b2 2)
se(b2)
t has a Student-t Distribution with T2 degrees of freedom.
5.16
Copyright 1996 Lawrence C. Marsh
Figure 5.1 Student-t Distribution
()
t0
f(t)
-tc tc
/2/2
red area = rejection region for 2-sided test
5.17
Copyright 1996 Lawrence C. Marsh
probability statements
P(-tc ٹ t ٹ tc) = 1
P( t < -tc ) = P( t > tc ) =
P(-tc ٹ tc) = 1 ٹ(b2 2)
se(b2)
5.18
Copyright 1996 Lawrence C. Marsh
Confidence Intervals
Two-sided (1)x100% C.I. for 1:
b1 t/2[se(b1)], b1 + t/2[se(b1)]
b2 t/2[se(b2)], b2 + t/2[se(b2)]
Two-sided (1)x100% C.I. for 2:
5.19
Copyright 1996 Lawrence C. Marsh
Student-t vs. Normal Distribution
1. Both are symmetric bell-shaped distributions.
2. Student-t distribution has fatter tails than the normal.
3. Student-t converges to the normal for infinite sample.
4. Student-t conditional on degrees of freedom (df).
5. Normal is a good approximation of Student-t for the first few decimal places when df > 30 or so.
5.20
Copyright 1996 Lawrence C. Marsh
Hypothesis Tests1. A null hypothesis, H0.
2. An alternative hypothesis, H1.
3. A test statistic.
4. A rejection region.
5.21
Copyright 1996 Lawrence C. Marsh
Rejection Rules1. Two-Sided Test:
If the value of the test statistic falls in the critical region in either tail of the t-distribution, then we reject the null hypothesis in favor of the alternative.
2. Left-Tail Test:If the value of the test statistic falls in the critical region which lies in the left tail of the t-distribution, then we reject the null hypothesis in favor of the alternative.
2. Right-Tail Test:If the value of the test statistic falls in the critical region which lies in the right tail of the t-distribution, then we reject the null hypothesis in favor of the alternative.
5.22
Copyright 1996 Lawrence C. Marsh
Format for Hypothesis Testing
1. Determine null and alternative hypotheses.
2. Specify the test statistic and its distribution as if the null hypothesis were true.
3. Select and determine the rejection region.
4. Calculate the sample value of test statistic.
5. State your conclusion.
5.23
Copyright 1996 Lawrence C. Marshpractical vs. statistical
significance in economics
Practically but not statistically significant:
When sample size is very small, a large average gap between the salaries of men and women might not be statistically significant.
Statistically but not practically significant:
When sample size is very large, a small correlation (say, = 0.00000001) between the winning numbers in the PowerBall Lottery and the Dow-Jones Stock Market Index might be statistically significant.
5.24
Copyright 1996 Lawrence C. Marsh
Type I and Type II errors
Type I error:
We make the mistake of rejecting the null hypothesis when it is true.
= P(rejecting H0 when it is true).
Type II error:
We make the mistake of failing to reject the null hypothesis when it is false. = P(failing to reject H0 when it is false).
5.25
Copyright 1996 Lawrence C. Marsh
Prediction Intervals
A (1)x100% prediction interval for yo is:
yo ± tc se( f )^
se( f ) = var( f )^f = yo yo^
x t x2var( f ) =
2 1 + +^ 1
x o x2^
5.26
Copyright 1996 Lawrence C. Marsh
The Simple Linear Regression Model
Chapter 6
Copyright © 1997 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein.
6.1
Copyright 1996 Lawrence C. Marsh
Explaining Variation in yt
Predicting yt without any explanatory variables:
yt = 1 + et
et = (yt 1)2
2
t = 1 t = 1
T T
= (yt b1) = 0et
2t = 1
t = 1
T
T
1
(yt b1) = 0t = 1
T
yt Tb1 = 0t = 1
T
b1 = y
Why not y?
6.2
Copyright 1996 Lawrence C. Marsh
Explaining Variation in yt
yt = b1 + b2xt + et^
Unexplained variation:
yt = b1 + b2xt^Explained variation:
et = yt yt = yt b1 b2xt^^
6.3
Copyright 1996 Lawrence C. Marsh
Explaining Variation in yt
yt = yt + et^
Why not y?
^
yt y = yt y + et^ ^
using y as baseline
SST = SSR + SSE(yty)2 = (yty)2 +et
t = 1
T ^ ^T T
t = 1 t = 1
2cross
producttermdropsout
6.4
Copyright 1996 Lawrence C. Marsh
Total Variation in yt
SST = total sum of squares
SST measures variation of yt around y
(yt y)2t = 1
T
SST =
6.5
Copyright 1996 Lawrence C. Marsh
Explained Variation in yt
SSR = regression sum of squares
yt = b1 + b2xt^Fitted yt values:^
SSR measures variation of yt around y^
(yt y)2t = 1
T
SSR = ^
6.6
Copyright 1996 Lawrence C. Marsh
Unexplained Variation in yt
SSE = error sum of squares
SSE measures variation of yt around yt^
et = ytyt = yt b1 b2xt^^
(yt yt)2 = et
2 t = 1
T
SSE = ^t = 1
T^
6.7
Copyright 1996 Lawrence C. Marsh
Analysis of Variance Table
^
Table 6.1 Analysis of Variance Table Source of Sum of MeanVariation DF Squares Square Explained 1 SSR SSR/1Unexplained T-2 SSE SSE/(T-2) [= 2]Total T-1 SST
6.8
Copyright 1996 Lawrence C. Marsh
Coefficient of Determination
1ٹ R2 ٹ 0
What proportion of the variation in yt is explained?
SSRSSTR2 =
6.9
Copyright 1996 Lawrence C. Marsh
Coefficient of Determination
SST = SSR + SSE
SST SSR SSESST SST SST
= +
SSR SSESST SST
1 = +
Dividingby SST
SSRSSTR2 = = 1 SSE
SST
6.10
Copyright 1996 Lawrence C. Marsh
R2 is only a descriptive measure.
R2 does not measure the qualityof the regression model.
Focusing solely on maximizing R2 is not a good idea.
Coefficient of Determination 6.11
Copyright 1996 Lawrence C. Marsh
cov(X,Y) =var(X) var(Y)
Correlation Analysis
cov(X,Y)r =var(X) var(Y)
Population:
^^ ^
Sample:
6.12
Copyright 1996 Lawrence C. Marsh
Correlation Analysis
var(X) =^(xt x)2/(T1)t = 1
T
var(Y) =^(yt y)2/(T1)t = 1
T
cov(X,Y) =^(xt x)(yt y)/(T1)t = 1
T
6.13
Copyright 1996 Lawrence C. Marsh
Correlation Analysis
(xt x)2 (yt y)2t = 1
T(xt x)(yt y)t = 1
T
r =t = 1
T
Sample Correlation Coefficient
6.14
Copyright 1996 Lawrence C. Marsh
Correlation Analysis and R2
For simple linear regression analysis:
r
2 = R2
R2 is also the correlation
between yt and yt
measuring “goodness of fit”.
^
6.15
Copyright 1996 Lawrence C. Marsh
Regression Computer Output
Table 6.2 Computer Generated Least Squares Results (1) (2) (3) (4) (5) Parameter Standard T for H0:Variable Estimate Error Parameter=0 Prob>|T|INTERCEPT 40.7676 22.1387 1.841 0.0734X 0.1283 0.0305 4.201 0.0002
Typical computer output of regression estimates:
6.16
Copyright 1996 Lawrence C. Marsh
Regression Computer Output
se(b1) = var(b1) = 490.12 = 22.1287^
se(b2) = var(b2) = 0.0009326 = 0.0305^
b1 = 40.7676 b2 = 0.1283
se(b1)t = = = 1.84
b1 40.7676
22.1287
se(b2)b2t = = = 4.20
0.12830.0305
6.17
Copyright 1996 Lawrence C. Marsh
Regression Computer Output
Table 6.3 Analysis of Variance Table Sum of MeanSource DF Squares Square Explained 1 25221.2229 25221.2229Unexplained 38 54311.3314 1429.2455Total 39 79532.5544 R-square: 0.3171
Sources of variation in the dependent variable:
6.18
Copyright 1996 Lawrence C. Marsh
Regression Computer Output
SSRSST
R2 = = 1 = 0.317SSESST
SSE /(T-2) = 2 = 1429.2455 ^SSE = et
2 = 54311^
SST = (yty)2 = 79532
SSR = (yty)2 = 25221^
6.19
Copyright 1996 Lawrence C. Marsh
yt = 40.7676 + 0.1283xt
(s.e.) (22.1387) (0.0305)
yt = 40.7676 + 0.1283xt
(t) (1.84) (4.20)
Reporting Regression Results 6.20
Copyright 1996 Lawrence C. Marsh
R2 = 0.317
Reporting Regression Results
This R2 value may seem low but it istypical in studies involving cross-sectionaldata analyzed at the individual or micro level.
A considerably higher R2 value would beexpected in studies involving time-series dataanalyzed at an aggregate or macro level.
6.21
Copyright 1996 Lawrence C. Marsh Effects of Scaling the Data
Changing the scale of x
yt = 1 + (c2)(xt/c) + et
yt = 1 + 2xt + et
yt = 1 + 2xt + et* *
2 = c2* xt = xt/c
*
where
and
The estimatedcoefficient andstandard errorchange but theother statisticsare unchanged.
6.22
Copyright 1996 Lawrence C. Marsh Effects of Scaling the Data
Changing the scale of y
yt/c = (1/c) + (2/c)xt + et/c
yt = 1 + 2xt + et
1 = 1/c * and
All statisticsare changedexcept forthe t-statisticsand R2 value.
yt = 1 + 2xt + et* ***
2 = 2/c *
*et = et/cyt = yt/cwhere *
6.23
Copyright 1996 Lawrence C. Marsh Effects of Scaling the Data
Changing the scale of x and y
yt/c = (1/c) + (c2/c)xt/c + et/c
yt = 1 + 2xt + et
1 = 1/c * and
No change inthe R2 or thet-statistics orin regression
results for 2
but all otherstats change.
yt = 1 + 2xt + et* ***
xt = xt/c *
*et = et/cyt = yt/cwhere *
6.24
Copyright 1996 Lawrence C. Marsh Functional Forms
The term linear in a simple regression model does not mean a linear relationship between variables, but a model in which the parameters enter the model in a linear way.
6.25
Copyright 1996 Lawrence C. Marsh
yt = 1 + 2xt + et
Linear Statistical Models:
Nonlinear Statistical Models:
ln(yt) = 1 + 2xt + et
yt = 1 + 2 ln(xt) + et
yt = 1 + 2xt + et2
yt = 1 + 2xt + et
3
yt = 1 + 2xt + exp(3xt) + et
yt = 1 + 2xt + et
3
Linear vs. Nonlinear 6.27
Copyright 1996 Lawrence C. Marsh
y
x
nonlinear relationship
between food expenditure and
income
Linear vs. Nonlinear
foodexpenditure
income0
6.27
Copyright 1996 Lawrence C. Marsh
Useful Functional Forms
1. Linear2. Reciprocal3. Log-Log4. Log-Linear5. Linear-Log6. Log-Inverse
Look at each form and its slope and elasticity
6.28
Copyright 1996 Lawrence C. Marsh
Linear
yt = 1 + 2xt + et
slope: 2 elasticity: 2 yt
Useful Functional Forms
xt
6.29
Copyright 1996 Lawrence C. Marsh
Reciprocal
yt = 1 + 2 + et
Useful Functional Forms
1
xtslope: elasticity:
1xt
221
xt yt2
6.30
Copyright 1996 Lawrence C. Marsh
xt
yt
Log-Log
ln(yt)= 1 + 2ln(xt) + et
slope: 2 elasticity: 2
Useful Functional Forms 6.31
Copyright 1996 Lawrence C. Marsh
Log-Linear
ln(yt)= 1 + 2xt + et
slope: 2 yt elasticity: 2xt
Useful Functional Forms 6.32
Copyright 1996 Lawrence C. Marsh
Linear-Log
yt= 1 + 2ln(xt) + et
_slope: 2 elasticity: 2
1xt
yt
1_
Useful Functional Forms 6.33
Copyright 1996 Lawrence C. Marsh
Useful Functional Forms
ln(yt) = 1 - 2 + et 1xt
Log-Inverse
slope: 2 elasticity: 2x2t
yt 1xt
6.34
Copyright 1996 Lawrence C. Marsh
1. E (et) = 0
2. var (et) = 2
3. cov(ei, ej) = 0
4. et ~ N(0, 2)
Error Term Properties 6.35
Copyright 1996 Lawrence C. Marsh
Economic Models
1. Demand Models2. Supply Models3. Production Functions4. Cost Functions5. Phillips Curve
6.36
Copyright 1996 Lawrence C. Marsh
1. Demand Models
* quality demanded (yd) and price (x)* constant elasticity
Economic Models
ln(yt )= 1 + 2ln(x)t + et d
6.37
Copyright 1996 Lawrence C. Marsh
2. Supply Models
* quality supplied (ys) and price (x)* constant elasticity
Economic Models
ln(yt )= 1 + 2ln(xt) + et s
6.38
Copyright 1996 Lawrence C. Marsh
3. Production Functions* output (y) and input (x)
* constant elasticity
Economic Models
ln(yt)= 1 + 2ln(xt) + et
Cobb-Douglas Production Function:
6.39
Copyright 1996 Lawrence C. Marsh
4a. Cost Functions
* total cost (y) and output (x)
Economic Models
yt = 1 + 2x2t + et
6.40
Copyright 1996 Lawrence C. Marsh
4b. Cost Functions
* average cost (x/y) and output (x)
Economic Models
(yt/xt) = 1/xt + 2xt + et/xt
6.41
Copyright 1996 Lawrence C. Marsh
5. Phillips Curve
* wage rate (wt) and time (t)
Economic Models
unemployment rate, ut
wt-1% wt =
wt wt-1= ut
1
nonlinear in both variables and parameters
6.42
Copyright 1996 Lawrence C. Marsh
The Multiple Regression Model
Chapter 7
Copyright © 1997 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein.
7.1
Copyright 1996 Lawrence C. MarshTwo Explanatory Variables
yt = 1 + 2xt2 + 3xt3 + et
yt
xt2
= 2 xt3
yt = 3xt‘s affect yt
separately
But least squares estimation of 2
now depends upon both xt2 and xt3 .
7.2
Copyright 1996 Lawrence C. MarshCorrelated Variables
yt = output xt2 = capital xt3 = labor
Always 5 workers per machine.
If number of workers per machine is never varied, it becomes impossible to tell if the machines or the workers are responsible for changes in output.
yt = 1 + 2xt2 + 3xt3 + et
7.3
Copyright 1996 Lawrence C. MarshThe General Model
yt = 1 + 2xt2 + 3xt3 +. . .+ KxtK + et
The parameter 1 is the intercept (constant) term.
The “variable” attached to 1 is xt1= 1.
Usually, the number of explanatory variables is said to be K1 (ignoring xt1= 1), while the
number of parameters is K. (Namely: 1 . . . K).
7.4
Copyright 1996 Lawrence C. Marsh
1. E(et) = 0
2. var(et) = 2
covet , es= for t ° s
4. et ~ N(0, 2)
Statistical Properties of et 7.5
Copyright 1996 Lawrence C. Marsh
1. E (yt) = 1 + 2xt2 +. . .+ KxtK
2. var(yt) = var(et) = 2
cov(yt ,ys) = cov(et , es) = 0 t°s
4. yt ~ N(1+2xt2 +. . .+KxtK, 2)
Statistical Properties of yt 7.6
Copyright 1996 Lawrence C. MarshAssumptions
1. yt = 1 + 2xt2 +. . .+ KxtK + et
2. E (yt) = 1 + 2xt2 +. . .+ KxtK
3. var(yt) = var(et) = 2
cov(yt ,ys) = cov(et ,es) = 0 t ° s
5. The values of xtk are not random
6. yt ~ N(1+2xt2 +. . .+KxtK, 2)
7.7
Copyright 1996 Lawrence C. Marsh
Least Squares Estimation
yt = 1 + 2xt2 + 3xt3 + et
S ؛ S(1, 2, 3) = yt12xt23xt3
t = 1
T
Define: yt = yt y*
xt2 = xt2 x2*
xt3 = xt3 x3*
7.8
Copyright 1996 Lawrence C. Marsh
b1 = yb1b2x2 b3x3
b3 = yt xt3xt2 yt xt2xt3xt2* * * * * * *2
xt2 xt3 xt2xt3* * * *2 2 2
b2 = yt xt2xt3 yt xt3xt2xt3* * * * * * *2
xt2 xt3 xt2xt3* * * *2 2 2
Least Squares Estimators 7.9
Copyright 1996 Lawrence C. Marsh
Dangers of Extrapolation
Statistical models generally are good only“within the relevant range”. This meansthat extending them to extreme data valuesoutside the range of the original data oftenleads to poor and sometimes ridiculous results.
If height is normally distributed and the normal ranges from minus infinity to plus infinity, pity the man minus three feet tall.
7.10
Copyright 1996 Lawrence C. MarshError Variance Estimation
2^ =
et^
Unbiased estimator of the error variance:
2
2
^
Transform to a chi-square distribution:
7.11
Copyright 1996 Lawrence C. MarshGauss-Markov Theorem
Under the assumptions of the multiple regression model, the ordinary least squares estimators have the smallest variance of all linear and unbiased estimators. This means that the least squares estimators are the Best Linear U nbiased Estimators (BLUE).
7.12
Copyright 1996 Lawrence C. Marsh
Variances
yt = 1 + 2xt2 + 3xt3 + et
var(b3) =
(1 r23)(xt3 x3)2
22
var(b2) =(1 r23)(xt2 x2)
222
(xt2 x2)2 (xt3 x3)
2
where r23 = (xt2 x2)(xt3 x3)
When r23 = 0these reduceto the simpleregressionformulas.
7.13
Copyright 1996 Lawrence C. Marsh
Variance Decomposition
The variance of an estimator is smaller when:
1. The error variance, 2, is smaller:
2 0 .
2. The sample size, T, is larger:
(xt2 x2)2 .
3. The variable’s values are more spread out: (xt2 x2)
2 .
4. The correlation is close to zero: r23 0 .
2
t = 1
T
7.14
Copyright 1996 Lawrence C. Marsh
Covariances
yt = 1 + 2xt2 + 3xt3 + et
where r23 =
(xt2 x2)2 (xt3 x3)
2
(xt2 x2)(xt3 x3)
(1 r23) (xt2 x2)2 (xt3 x3)
2
cov(b2,b3) = 2
r23 2
7.15
Copyright 1996 Lawrence C. Marsh
Covariance Decomposition
1. The error variance, 2, is larger.
2. The sample size, T, is smaller.
3. The values of the variables are less spread out.
4. The correlation, r23, is high.
The covariance between any two estimatorsis larger in absolute value when:
7.16
Copyright 1996 Lawrence C. MarshVar-Cov Matrix
yt = 1 + 2xt2 + 3xt3 + et
var(b1) cov(b1,b2) cov(b1,b3)cov(b1,b2,b3) = cov(b1,b2) var(b2) cov(b2,b3) cov(b1,b3) cov(b2,b3) var(b3)
The least squares estimators b1, b2, and b3
have covariance matrix:
7.17
Copyright 1996 Lawrence C. MarshNormal
yt = 1 + 2x2t + 3x3t +. . .+ KxKt + et
yt ~N (1 + 2x2t + 3x3t +. . .+ KxKt), 2
et ~ N(0, 2)This implies and is implied by:
bk ~ N k, var(bk)
z = ~ N(0,1) for k = 1,2,...,Kbk k
var(bk)
Since bk is a linear
function of the yt’s:
7.18
Copyright 1996 Lawrence C. MarshStudent-t
bk k
var(bk)^t = =
bk k
se(bk)
Since generally the population varianceof bk , var(bk) , is unknown, we estimate it with which uses
2 instead of 2.var(bk)^ ^
t has a Student-t distribution with df=(TK).
7.19
Copyright 1996 Lawrence C. Marsh
Interval Estimation
bk k
se(bk)P tc ٹ tc = 1 ٹ
tc is critical value for (T-K) degrees of freedom
such that P(t چ tc) = /2.
P bk tc se(bk) ٹ k ٹ bk + tc se(bk) = 1
Interval endpoints: bk tc se(bk) , bk + tc se(bk)
7.20
Copyright 1996 Lawrence C. Marsh
Hypothesis Testing and
Nonsample Information
Chapter 8
Copyright © 1997 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein.
8.1
Copyright 1996 Lawrence C. Marsh
1. Student-t Tests
2. Goodness-of-Fit
3. F-Tests
4. ANOVA Table
5. Nonsample Information
6. Collinearity
7. Prediction
Chapter 8: Overview 8.2
Copyright 1996 Lawrence C. MarshStudent - t Test
yt = 1 + 2Xt2 + 3Xt3 + 4Xt4 + et
Student-t tests can be used to test any linearcombination of the regression coefficients:
H0: 2 + 3 + 4 = 1H0: 1 = 0
H0: 32 73 = 21 H0: 2 3 5ٹ
Every such t-test has exactly TK degrees of freedomwhere K=#coefficients estimated(including the intercept).
8.3
Copyright 1996 Lawrence C. MarshOne Tail Test
yt = 1 + 2Xt2 + 3Xt3 + 4Xt4 + et
H0: 3 0ٹ
H1: 3 > 0b3
se(b3)t = ~ t (TK)
tc0
df = TK = T4
8.4
Copyright 1996 Lawrence C. MarshTwo Tail Test
yt = 1 + 2Xt2 + 3Xt3 + 4Xt4 + et
H0: 2 = 0
H1: 2 ° 0b2
se(b2)t = ~ t (TK)
tc0
df = TK = T4
-tc
8.5
Copyright 1996 Lawrence C. MarshGoodness - of - Fit
1ٹ R2 ٹ 0
Coefficient of Determination
SSTR2 = = (yt y)2t = 1
T^
SSR
(yt y)2t = 1
T
8.6
Copyright 1996 Lawrence C. MarshAdjusted R-Squared
Adjusted Coefficient of Determination
Original:
Adjusted:
SST/(T1)R2 = 1 SSE/(TK)
SST= 1 SSER2 = SSTSSR
8.7
Copyright 1996 Lawrence C. MarshComputer Output
Table 8.2 Summary of Least Squares Results Variable Coefficient Std Error t-value p-value constant 104.79 6.48 16.17 0.000price 6.642 3.191 2.081 0.042advertising 2.984 0.167 17.868 0.000
b2
se(b2)t = =
6.642
3.1912.081=
8.8
Copyright 1996 Lawrence C. MarshReporting Your Results
yt = Xt2 + Xt3^
(6.48) (3.191) (0.167) (s.e.)
yt = Xt2 + Xt3^
(16.17) (-2.081) (17.868) (t)
Reporting t-statistics:
Reporting standard errors:
8.9
Copyright 1996 Lawrence C. MarshSingle Restriction F-Test
yt = 1 + 2Xt2 + 3Xt3 + 4Xt4 + et
H0: 2 = 0
H1: 2 ° 0
dfd = TK = 49dfn = J = 1
(SSER SSEU)/J
SSEU/(TK)F =
(1964.758 1805.168)/1
1805.168/(52 3)=
= 4.33
By definition this is the t-statistic squared:t = 2.081 F = t2 =
8.10
Copyright 1996 Lawrence C. MarshMultiple Restriction F-Test
yt = 1 + 2Xt2 + 3Xt3 + 4Xt4 + et
H0: 2 = 0, 4 = 0
H1: H0 not true
dfd = TK = 49dfn = J = 2
(SSER SSEU)/J
SSEU/(TK)F =
First run the restrictedregression by dropping Xt2 and Xt4 to get SSER.Next run unrestricted regression to get SSEU .
8.11
Copyright 1996 Lawrence C. Marsh
F-Tests
(SSER SSEU)/J
SSEU/(TK)F =
F-Tests of this type are always right-tailed, even for left-sided or two-sided hypotheses, because any deviation from the null will make the F value bigger (move rightward).
0 Fc
f(F)
F
8.12
Copyright 1996 Lawrence C. MarshF-Test of Entire Equation
yt = 1 + 2Xt2 + 3Xt3 + et
H0: 2 = 3 = 0
H1: H0 not true
dfd = TK = 49dfn = J = 2
(SSER SSEU)/J
SSEU/(TK)F =
(13581.35 1805.168)/2
1805.168/(52 3)=
= 159.828
We ignore 1. Why?
Fc = 3.187
= 0.05
Reject H0!
8.13
Copyright 1996 Lawrence C. MarshANOVA Table
Table 8.3 Analysis of Variance Table Sum of MeanSource DF Squares Square F-Value Explained 2 11776.18 5888.09 158.828Unexplained 49 1805.168 36.84Total 51 13581.35 p-value: 0.0001
SSTR2 = =SSR = 0.86711776.18
13581.35
8.14
Copyright 1996 Lawrence C. MarshNonsample Information
ln(yt) = 1 + 2 ln(Xt2) + 3 ln(Xt3) + 4 ln(Xt4) + et
A certain production process is known to beCobb-Douglas with constant returns to scale.
2 + 3 + 4 = 1where 4 = (1 2 3)
ln(yt /Xt4) = 1 + 2 ln(Xt2/Xt4) + 3 ln(Xt3 /Xt4) + et
yt = 1 + 2 Xt2 + 3 Xt3 + 4 Xt4 + et* * * *
Run least squares on the transformed model.Interpret coefficients same as in original model.
8.15
Copyright 1996 Lawrence C. MarshCollinear Variables
The term “independent variable” means an explanatory variable is independent of of the error term, but not necessarily independent of other explanatory variables.
Since economists typically have no controlover the implicit “experimental design”,explanatory variables tend to movetogether which often makes sorting outtheir separate influences rather problematic.
8.16
Copyright 1996 Lawrence C. MarshEffects of Collinearity
1. no least squares output when collinearity is exact.
2. large standard errors and wide confidence intervals.
3. insignificant t-values even with high R2 and a significant F-value.
4. estimates sensitive to deletion or addition of a few observations or “insignificant” variables.
5. good “within-sample”(same proportions) but poor “out-of-sample”(different proportions) prediction.
A high degree of collinearity will produce:
8.17
Copyright 1996 Lawrence C. MarshIdentifying Collinearity
Evidence of high collinearity include:
1. a high pairwise correlation between two explanatory variables.
2. a high R-squared when regressing one explanatory variable at a time on each of the remaining explanatory variables.
3. a statistically significant F-value when the t-values are statistically insignificant.
4. an R-squared that doesn’t fall by much when dropping any of the explanatory variables.
8.18
Copyright 1996 Lawrence C. MarshMitigating Collinearity
Since high collinearity is not a violation ofany least squares assumption, but rather a lack of adequate information in the sample:
1. collect more data with better information.
2. impose economic restrictions as appropriate.
3. impose statistical restrictions when justified.
4. if all else fails at least point out that the poor model performance might be due to the collinearity problem (or it might not).
8.19
Copyright 1996 Lawrence C. MarshPrediction
Given a set of values for the explanatoryvariables, (1 X02 X03), the best linearunbiased predictor of y is given by:
yt = 1 + 2Xt2 + 3Xt3 + et
This predictor is unbiased in the sensethat the average value of the forecasterror is zero.
y0 = b1 + b2X02 + b3X03^
8.20
Copyright 1996 Lawrence C. Marsh
Extensions of the Multiple
Regression Model
Chapter 9
Copyright © 1997 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein.
9.1
Copyright 1996 Lawrence C. Marsh
Topics for This Chapter
1. Intercept Dummy Variables
2. Slope Dummy Variables
3. Different Intercepts & Slopes
4. Testing Qualitative Effects
5. Are Two Regressions Equal?
6. Interaction Effects
7. Dummy Dependent Variables
9.2
Copyright 1996 Lawrence C. Marsh
Intercept Dummy Variables
Dummy variables are binary (0,1)
Dt = 1 if red car, Dt = 0 otherwise.
yt = 1 + 2Xt + 3Dt + et
yt = speed of car in miles per hour
Xt = age of car in years
Police: red cars travel faster.H0: 3 = 0H1: 3 > 0
9.3
Copyright 1996 Lawrence C. Marsh
yt = 1 + 2Xt + 3Dt + et
red cars: yt = (1 + 3) + 2xt + et other cars: yt = 1 + 2Xt + et
yt
Xt
milesper hour
age in years0
1 + 3
12
2
red cars
other cars
9.4
Copyright 1996 Lawrence C. MarshSlope Dummy Variables
yt = 1 + 2Xt + 3DtXt + et
yt = 1 + (2 + 3)Xt + et
yt = 1 + 2Xt + et
yt
Xt
valueofporfolio
years0
2 + 3
1
2
stocks
bonds
Stock portfolio: Dt = 1 Bond portfolio: Dt = 0
1 = initial
investment
9.5
Copyright 1996 Lawrence C. Marsh
Different Intercepts & Slopes
yt = 1 + 2Xt + 3Dt + 4DtXt + et
yt = (1 + 3) + (2 + 4)Xt + et
yt = 1 + 2Xt + et
yt
Xt
harvestweightof corn
rainfall
2 + 4
1
2
“miracle”
regular
“miracle” seed: Dt = 1 regular seed: Dt = 0
1 + 3
9.6
Copyright 1996 Lawrence C. Marshyt = 1 + 2 Xt + 3 Dt + et
21+ 3
2
1
yt
Xt
Men
Women
0
yt = 1 + 2 Xt + et
For men Dt = 1. For women Dt = 0.
years of experience
yt = (1+ 3) + 2 Xt + et
wagerate
H0: 3 = 0
H1: 3 > 0 .
. Testing fordiscriminationin starting wage
9.7
Copyright 1996 Lawrence C. Marshyt = 1 + 5 Xt + 6 Dt Xt + et
5
5 +6
1
yt
Xt
Men
Women
0
yt = 1 + (5 +6 )Xt + et
yt = 1 + 5 Xt + et
For men Dt = 1.
For women Dt = 0.
Men and women have the same starting wage, 1 , but their wage ratesincrease at different rates (diff.= 6 ).
6 > means that men’s wage rates areincreasing faster than women's wage rates.
years of experience
wagerate
9.8
Copyright 1996 Lawrence C. Marsh
yt = 1 + 2 Xt + 3 Dt + 4 Dt Xt + et
1 + 3
1
2
2 + 4
yt
Xt
Men
Women
0
yt = (1 + 3) + (2 + 4) Xt + et
yt = 1 + 2 Xt + et
Women are given a higher starting wage, 1 , while men get the lower starting wage, 1 + 3 ,(3 < 0 ). But, men get a faster rate of increasein their wages, 2 + 4 , which is higher than therate of increase for women, 2 , (since 4 > 0 ).
years of experience
An Ineffective Affirmative Action Plan
women are startedat a higher wage.
Note:(3 < 0 )
wagerate
9.9
Copyright 1996 Lawrence C. Marsh
Testing Qualitative Effects
1. Test for differences in intercept.
2. Test for differences in slope.
3. Test for differences in both intercept and slope.
9.10
Copyright 1996 Lawrence C. Marsh
H0: vs1:
H0: vs1:
Yt 12Xt3Dt
4Dt Xt
b3
Est. Var b3کt n 4
b4
Est. Var b4کt n 4
men: Dt = 1 ; women: Dt = 0
Testing fordiscrimination instarting wage.
Testing fordiscrimination inwage increases.
intercept
slope
et
9.11
Copyright 1996 Lawrence C. Marsh
Testing Ho:
H1 : otherwise
and
SSE R yt b 1 b 2 X t 2
t 1
T
ه
SSE U yt b1bXt bDt b Dt Xt2
t1
T
ه
SSER SSEU 2
SSEU T 4 F T 4
intercept and slope
9.12
Copyright 1996 Lawrence C. Marsh
Are Two Regressions Equal?
yt = 1 + 2 Xt + 3 Dt + 4 Dt Xt + et
variations of “The Chow Test”
I. Assuming equal variances (pooling):
men: Dt = 1 ; women: Dt = 0
Ho: 3 = 4 = 0 vs. H1: otherwise
yt = wage rate
This model assumes equal wage rate variance.
Xt = years of experience
9.13
Copyright 1996 Lawrence C. Marsh
yt = 1 + 2 Xt + et
II. Allowing for unequal variances:
ytm = 1 + 2 Xtm + etm
ytw = 1 + 2 Xtw + etw
Everyone:
Men only:Women only:
SSER
Forcing men and women to have same 1, 2.
Allowing men and women to be different.SSEm
SSEw
where SSEU = SSEm + SSEw
F =(SSER SSEU)/J
SSEU /(TK)
J = # restrictions
K=unrestricted coefs.
(running three regressions)
J = 2 K = 4
9.14
Copyright 1996 Lawrence C. Marsh
Interaction Variables
1. Interaction Dummies
2. Polynomial Terms (special case of continuous interaction)
3. Interaction Among Continuous Variables
9.15
Copyright 1996 Lawrence C. Marsh
1. Interaction Dummies
yt = 1 + 2 Xt + 3 Mt + 4 Bt + et
For men Mt = 1. For women Mt = 0. For black Bt = 1. For nonblack Bt = 0.
No Interaction: wage gap assumed the same:
yt = 1 + 2 Xt + 3 Mt + 4 Bt + 5 Mt Bt + et
Interaction: wage gap depends on race:
Wage Gap between Men and Women
yt = wage rate; Xt = experience
9.16
Copyright 1996 Lawrence C. Marsh
2. Polynomial Terms
yt = 1 + 2 X t + 3 X2
t + 4 X3
t + et
Linear in parameters but nonlinear in variables:
yt = income; Xt = agePolynomial Regression
yt
X tPeople retire at different ages or not at all.
9020 30 40 50 60 8070
9.17
Copyright 1996 Lawrence C. Marsh
yt = 1 + 2 X t + 3 X2
t + 4 X3
t + et
yt = income; Xt = age
Polynomial Regression
Rate income is changing as we age:yt
Xt
= 2 + 2 3 X t + 3 4 X
2t
Slope changes as X t changes.
9.18
Copyright 1996 Lawrence C. Marsh
3. Continuous Interaction
yt = 1 + 2 Zt + 3 Bt + 4 Zt Bt + et
Exam grade = f(sleep:Zt , study time:Bt)
Sleep and study time do not act independently.
More study time will be more effective when combined with more sleep and less effective when combined with less sleep.
9.19
Copyright 1996 Lawrence C. Marsh
Your mind sortsthings out whileyou sleep (when you have things to sort out.)
yt = 1 + 2 Zt + 3 Bt + 4 Zt Bt + et
Exam grade = f(sleep:Zt , study time:Bt)
yt
Bt
= 2 + 4 Zt
Your studying is more effectivewith more sleep.
yt
Zt
= 2 + 4 Bt
continuous interaction 9.20
Copyright 1996 Lawrence C. Marsh
yt = 1 + 2 Zt + 3 Bt + 4 Zt Bt + et
Exam grade = f(sleep:Zt , study time:Bt)
If Zt + Bt = 24 hours, then Bt = (24 Zt)
yt = 1+ 2 Zt +3(24 Zt) +4 Zt (24 Zt) + et
yt = (1+24 3) + (23+24 4)Zt 4Z2
t + et
yt = 1 + 2 Zt + 3 Z2
t + et
Sleep needed to maximize your exam grade:yt
Zt
= 2 + 23 Zt = 0where 2 > 0 and 3 < 0
2
3
Zt =
9.21
Copyright 1996 Lawrence C. Marsh
1. Linear Probability Model
2. Probit Model
3. Logit Model
Dummy Dependent Variables9.22
Copyright 1996 Lawrence C. Marsh
Linear Probability Model
yi = 1 + 2 Xi2 + 3 Xi3 + 4 Xi4 + ei
Xi2 = total hours of work each week
1 quits job 0 does not quit
yi =
Xi3 = weekly paycheck
Xi4 = hourly pay (Xi3 divided by Xi2)
9.23
Copyright 1996 Lawrence C. Marsh
Xi2
yi = 1 + 2 Xi2 + 3 Xi3 + 4 Xi4 + ei
yt = 1
0yt =total hours of work each week
yi = b1 + b2 Xi2 + b3 Xi3 + b4 Xi4^
yi^
Read predicted values of yi off the regression line
Linear Probability Model 9.24
Copyright 1996 Lawrence C. Marsh
1. Probability estimates are sometimesless than zero or greater than one.
2. Heteroskedasticity is present in that the model generates a nonconstant error variance.
Linear Probability Model
Problems with Linear Probability Model:
9.25
Copyright 1996 Lawrence C. Marsh
Probit Model
zi = 1 + 2 Xi2 +
2 f(zi) = e0.5zi
21
F(zi) = P[ Z zi ] = e0.5u2
du 2 1
Normal probability density function:
Normal cumulative probability function:
zi
latent variable, zi :
9.26
Copyright 1996 Lawrence C. Marsh
pi = P[ Z 1 + 2Xi2 ] = F(1 + 2Xi2)
Since zi = 1 + 2 Xi2 + , we cansubstitute in to get
Probit Model
Xi2total hours of work each week
yt = 1
0yt =
9.27
Copyright 1996 Lawrence C. Marsh
Logit Model
pi =1
1 + e (1 +
2 X
i2 +
Define pi :
For 2 > 0, pi will approach 1 as Xi2 +
pi is the probability of quitting the job.
For 2 > 0, pi will approach 0 as Xi2
9.28
Copyright 1996 Lawrence C. Marsh
Logit Model
Xi2total hours of work each week
yt = 1
0yt =
pi =1
1 + e (1 +
2 X
i2 +
pi is the probability of quitting the job.
9.29
Copyright 1996 Lawrence C. Marsh
Maximum Likelihood
Maximum likelihood estimation (MLE)is used to estimate Probit and Logit functions.
The small sample properties of MLE are not known, but in large samples MLE is normally distributed, and it is consistent and asymptotically efficient.
9.30
Copyright 1996 Lawrence C. Marsh
Heteroskedasticity
Chapter 10
Copyright © 1997 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein.
10.1
Copyright 1996 Lawrence C. Marsh
The Nature of Heteroskedasticity
Heteroskedasticity is a systematic pattern in the errors where the variances of the errors are not constant.
Ordinary least squares assumes that all observations are equally reliable.
For efficiency (accurate estimation/prediction) reweight observations to ensure equal error variance.
10.2
Copyright 1996 Lawrence C. Marsh
yt = 1 + 2xt + et
Regression Model
E(et) = 0
var(et) = 2
zero mean:
homoskedasticity:
nonautocorrelation: cov(et, es) = t ° s
heteroskedasticity: var(et) = t2
10.3
Copyright 1996 Lawrence C. Marsh
Homoskedastic pattern of errors
xt
yt
...
....
...
...
..
..
. .. ... . .
. .. .. .
. ...
...
..
.
income
consumption
10.4
Copyright 1996 Lawrence C. Marsh
..
xtx1 x2
y tf(yt)
The Homoskedastic Case
..
x3 x4 income
cons
umpti
on
10.5
Copyright 1996 Lawrence C. Marsh
Heteroskedastic pattern of errors
xt
yt
.
... .
. .. .
..
.
.
.
.
..
..
.
..
.
..
...
.
..
..
..... ..... .
..
income
consumption
10.6
Copyright 1996 Lawrence C. Marsh
.
x tx1 x2
y tf(yt)
cons
umpti
on
x3
..
The Heteroskedastic Case
income
rich people
poor people
10.7
Copyright 1996 Lawrence C. Marsh
Properties of Least Squares
1. Least squares still linear and unbiased.
2. Least squares not efficient.
3. Usual formulas give incorrect standard errors for least squares.
4. Confidence intervals and hypothesis tests based on usual standard errors are wrong.
10.8
Copyright 1996 Lawrence C. Marsh
yt = 1 + 2xt + et
heteroskedasticity: var(et) = t2
incorrect formula for least squares variance:
var(b2) = 2
xt x
correct formula for least squares variance:
var(b2) = t2xt x
xt x
10.9
Copyright 1996 Lawrence C. Marsh
Hal White’s Standard Errors
White’s estimator of the least squares variance:
est.var(b2) = et2xt x
xt x
^
In large samples White’s standard error (square root of estimated variance) is acorrect / accurate / consistent measure.
10.10
Copyright 1996 Lawrence C. Marsh
Two Types of Heteroskedasticity
1. Proportional Heteroskedasticity. (continuous function(of xt, for example))
2. Partitioned Heteroskedasticity. (discrete categories/groups)
10.11
Copyright 1996 Lawrence C. Marsh
Proportional Heteroskedasticity
yt = 1 + 2xt + et
where
var(et) = t2E(et) = 0 cov(et, es) = 0 t ° s
t2 = 2 xt
The variance is assumed to be proportional to the value of xt
10.12
Copyright 1996 Lawrence C. Marsh
t2 = 2 xt
yt = 1 + 2xt + et
std.dev. proportional to xt
variance:
standard deviation: t = xt
yt 1 xt et= 1 + 2 + xt xt xt xt
To correct for heteroskedasticity divide the model by xt
var(et) = t2
10.13
Copyright 1996 Lawrence C. Marshyt 1 xt et= 1 + 2 + xt xt xt xt
yt = 1xt1 + 2xt2 + et* * * *
var(et ) = var( ) = var(et) = 2 xt* et
xt
1xt
1xt
var(et ) = 2*
et is heteroskedastic, but et is homoskedastic.*
10.14
Copyright 1996 Lawrence C. Marsh
1. Decide which variable is proportional to the heteroskedasticity (xt in previous example).
2. Divide all terms in the original model by the square root of that variable (divide by xt ).
3. Run least squares on the transformed model which has new yt, xt1 and xt2 variables
but no intercept.
Generalized Least Squares
These steps describe weighted least squares:
* * *
10.15
Copyright 1996 Lawrence C. Marsh
Partitioned Heteroskedasticity
yt = 1 + 2xt + et
var(et) = 12
var(et) = 22
error variance of “field” corn:
error variance of “sweet” corn:
yt = bushels per acre of corn
xt = gallons of water per acre (rain or other)
t = 1, . . . ,100
t = 1, . . . ,80
t = 81, . . . ,100
10.16
Copyright 1996 Lawrence C. Marsh
yt = 1 + 2xt + et var(et) = 12“field” corn:
yt = 1 + 2xt + et var(et) = 22“sweet” corn:
yt 1 xt et= 1 + 2 + 1 1 1 1
yt 1 xt et= 1 + 2 + 2 2 2 2
Reweighting Each Group’s Observations
t = 1, . . . ,80
t = 81, . . . ,100
10.17
Copyright 1996 Lawrence C. Marsh
Apply Generalized Least Squares
Run least squares separately on data for each group.
12 provides estimator of 1
2 using
the 80 observations on “field” corn.
^
22 provides estimator of 2
2 using
the 20 observations on “sweet” corn.
^
10.18
Copyright 1996 Lawrence C. Marsh
1. Residual Plots provide information on the exact nature of heteroskedasticity (partitioned or proportional) to aid in correcting for it.
2. Goldfeld-Quandt Test checks for presence of heteroskedasticity.
Detecting Heteroskedasticity
Determine existence and nature of heteroskedasticity:
10.19
Copyright 1996 Lawrence C. Marsh
Residual Plots
et
0
xt
.
.
..
.
.
.. .
..
..
..
.
... .
..
..
. .. ..
.
. . ..
.....
..
..
.
Plot residuals against one variable at a timeafter sorting the data by that variable to tryto find a heteroskedastic pattern in the data.
10.20
Copyright 1996 Lawrence C. Marsh
Goldfeld-Quandt Test
The Goldfeld-Quandt test can be used to detect heteroskedasticity in either the proportional case or for comparing two groups in the discrete case.
For proportional heteroskedasticity, it is first necessaryto determine which variable, such as xt, is proportionalto the error variance. Then sort the data from the largest to smallest values of that variable.
10.21
Copyright 1996 Lawrence C. Marsh
Ho: 12 = 2
2
H1: 12 > 2
2
GQ = ~ F[T1-K1, T2-K2]
12
22
^
^
In the proportional case, drop the middle r observations where r T/6, then runseparate least squares regressions on the firstT1 observations and the last T2 observations.
Small values of GQ support Ho while large values support H1.
Goldfeld-Quandt Test Statistic
Use F Table
10.22
Copyright 1996 Lawrence C. Marsh
t2 = 2 exp{1 zt1 + 2 zt2}
More General Model
Structure of heteroskedasticity could be more complicated:
zt1 and zt2 are any observable variables upon
which we believe the variance could depend.
Note: The function exp{.} ensures that t2 is positive.
10.23
Copyright 1996 Lawrence C. Marsh
t2 = 2 exp{1 zt1 + 2 zt2}
More General Model
lnt2 = ln
2+ 1 zt1 + 2 zt2
lnt2 = + 1 zt1 + 2 zt2
where = ln2
Ho: 1 = 0, 2
= 0
H1: 1 ° 0, 2
° 0 and/or
Least squares residuals, et ^
lnet2 =+1zt1+2zt2 + t
^
the usual F test
10.24
Copyright 1996 Lawrence C. Marsh
Autocorrelation
Chapter 11
Copyright © 1997 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein.
11.1
Copyright 1996 Lawrence C. Marsh
The Nature of Autocorrelation
For efficiency (accurate estimation/prediction) all systematic information needs to be incor-porated into the regression model.
Autocorrelation is a systematic pattern in the errors that can be either attracting (positive) or repelling (negative) autocorrelation.
11.2
Copyright 1996 Lawrence C. Marsh
PostiveAuto.
NoAuto.
NegativeAuto.
et
.0
et
0
et
0
t
t
t
.. . . .
. . . . .. .
. . . ... ...
.. .
.
.. ... .
...
.. .. ... .
....
..
.
.
..
.
...
.
..
. .
....
.
.
crosses line not enough (attracting)
crosses line randomly
crosses line too much (repelling)
11.3
Copyright 1996 Lawrence C. Marsh
yt = 1 + 2xt + et
Regression Model
E(et) = 0
var(et) = 2
zero mean:
homoskedasticity:
nonautocorrelation: cov(et, es) = t ° s
autocorrelation: cov(et, es) ° t ° s
11.4
Copyright 1996 Lawrence C. Marsh
Order of Autocorrelation
yt = 1 + 2xt + et
et = et1 + t
et = 1et1 + 2et2 + t
et = 1et1 + 2et2 + 3et3 + t
1st Order:
2nd Order:
3rd Order:
We will assume First Order Autocorrelation:
et = et1 + tAR(1) :
11.5
Copyright 1996 Lawrence C. Marsh
First Order Autocorrelation
yt = 1 + 2xt + et
et = et1 + t where 1 < < 1
E(t) = 0 var(t) = 2 cov(t, s) = t ° s
These assumptions about t imply the following about et :
E(et) = 0
var(et) = e2 =
cov(et, etk) = e2 k for k > 0
corr(et, etk) = k for k > 0
2
12
11.6
Copyright 1996 Lawrence C. Marsh
Autocorrelation creates someProblems for Least Squares:
1. The least squares estimator is still linear and unbiased but it is not efficient.
2. The formulas normally used to compute the least squares standard errors are no longer correct and confidence intervals and hypothesis tests using them will be wrong.
11.7
Copyright 1996 Lawrence C. Marsh
Generalized Least Squares
yt = 1 + 2xt + et
et = et1 + t
yt = 1 + 2xt + et1 + t
substitute in for et
Now we need to get rid of et1
(continued)
AR(1) :
11.8
Copyright 1996 Lawrence C. Marsh
yt = 1 + 2xt + et
yt = 1 + 2xt + et1 + t
et = yt 12xt
et1 = yt1 12xt1
yt = 1 + 2xt + yt1 12xt1 + t
lag theerrorsonce
(continued)
11.9
Copyright 1996 Lawrence C. Marsh
yt = 1 + 2xt + yt1 12xt1 + t
yt = 1 + 2xt + yt1 12xt1 + t
yt yt1 = 1(1) + 2(xtxt1) + t
yt = 1 + 2xt2 + t* * *
yt = yt yt1 *
1 = 1(1) *
xt2 = (xtxt1) *
11.10
Copyright 1996 Lawrence C. Marsh
yt = 1 + 2xt2 + t* * *
yt = yt yt1 * 1 = 1(1) *
xt2 = xt xt1*
Problems estimating this model with least squares:
1. One observation is used up in creating the transformed (lagged) variables leaving only (T1) observations for estimating the model.
2. The value of is not known. We must find some way to estimate it.
11.11
Copyright 1996 Lawrence C. Marsh
Recovering the 1st Observation
Dropping the 1st observation and applying least squares is not the best linear unbiased estimation method.
Efficiency is lost because the varianceof the error associated with the 1st observation is not equal to that of the other errors.
This is a special case of the heteroskedasticityproblem except that here all errors are assumedto have equal variance except the 1st error.
11.12
Copyright 1996 Lawrence C. Marsh
Recovering the 1st Observation
y1 = 1 + 2x1 + e1
The 1st observation should fit the original model as:
We could include this as the 1st observation for ourestimation procedure but we must first transform it sothat it has the same error variance as the other observations.
with error variance: var(e1) = e2 =
2 /(1-2).
Note: The other observations all have error variance 2.
11.13
Copyright 1996 Lawrence C. Marsh
y1 = 1 + 2x1 + e1
with error variance: var(e1) = e2 =
2 /(1-2).
The other observations all have error variance 2.
Given any constant c : var(ce1) = c2 var(e1).
If c = 1-2 , then var( 1-2 e1) = (1-2) var(e1).
= (1-2) e2
= (1-2) 2 /(1-2)
= 2
The transformation 1 = 1-2 e1 has variance 2 .
11.14
Copyright 1996 Lawrence C. Marsh
y1 = 1 + 2x1 + e1
The transformed error 1 = 1-2 e1 has variance 2 .
Multiply through by 1-2 to get:
1-2 y1 = 1-2 1 + 1-2 2x1 + 1-2 e1
This transformed first observation may now be added to the other (T-1) observations to obtain the fully restored set of T observations.
11.15
Copyright 1996 Lawrence C. Marsh
Estimating Unknown Value
et = et1 + t
First, use least squares to estimate the model:
If we had values for the et’s, we could estimate:
yt = 1 + 2xt + et
The residuals from this estimation are:
et = yt - b1 - b2xt^
11.16
Copyright 1996 Lawrence C. Marsh
et = yt - b1 - b2xt^
et = et1 + t^ ^ ^
Next, estimate the following by least squares:
The least squares solution is:
et et-1
et-1
T
T
t = 2
t = 2
2
^ ^
^=^
11.17
Copyright 1996 Lawrence C. Marsh
Durbin-Watson Test
Ho: = 0 vs. H1: ° 0 , > 0, or < 0
et et-1
et
T
T
t = 2
t = 1
2
^ ^
^d=
2
The Durbin-Watson Test statistic, d, is :
11.18
Copyright 1996 Lawrence C. Marsh
Testing for AutocorrelationThe test statistic, d, is approximately related to as:^
d 2(1)^
When = 0 , the Durbin-Watson statistic is d 2.^
When = 1 , the Durbin-Watson statistic is d 0.^
Tables for critical values for d are not always readily available so it is easier to use the p-valuethat most computer programs provide for d.
Reject Ho if p-value < , the significance level.
11.19
Copyright 1996 Lawrence C. Marsh
Prediction with AR(1) Errors
When errors are autocorrelated, the previous period’s error may help us predict next period’s error.
The best predictor, yT+1 , for next period is:
yT+1 = 1 + 2xT+1 + eT^ ^ ^ ^ ~
where 1 and 2 are generalized least squares
estimates and eT is given by:~
^ ^
eT = yT 1 2xT ^ ^ ~
11.20
Copyright 1996 Lawrence C. Marsh
yT+h = 1 + 2xT+h + h eT^ ^ ^ ^ ~
For h periods ahead, the best predictor is:
Assuming | | < 1, the influence of h eT
diminishes the further we go into the future(the larger h becomes).
^ ^ ~
11.21
Copyright 1996 Lawrence C. Marsh
Pooling
Time-Series and
Cross-Sectional Data
Chapter 12
Copyright © 1997 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein.
12.1
Copyright 1996 Lawrence C. Marsh
Pooling Time and Cross Sections
yit = 1it + 2itx2it + 3itx3it + eit
If left unrestricted,this model requires different equationsfor each firm in each time period.
for the ith firm in the tth time period
12.2
Copyright 1996 Lawrence C. Marsh
Seemingly Unrelated Regressions
yit = 1i + 2ix2it + 3ix3it + eit
SUR models impose the restrictions:
1it = 1i 2it = 2i 3it = 3i
Each firm gets its own coefficients: 1i , 2i and 3i
but those coefficients are constant over time.
12.3
Copyright 1996 Lawrence C. Marsh
The investment expenditures (INV) of General Electric (G) and Westinghouse(W) may be related to their stock marketvalue (V) and actual capital stock (K) as follows:
INVGt = 1G + 2GVGt + 3GKGt + eGt
INVWt = 1W + 2WVWt + 3WKWt + eWt
i = G, W t = 1, . . . , 20
Two-Equation SUR Model12.4
Copyright 1996 Lawrence C. Marsh
Estimating Separate Equations
For now make the assumption of no correlationbetween the error terms across equations:
We make the usual error term assumptions:
cov(eGt, eGs) = 0 cov(eWt, eWs) = 0
var(eGt) = G2
var(eWt) = W2
E(eGt) = 0 E(eWt) = 0
cov(eGt, eWt) = 0 cov(eGt, eWs) = 0
12.5
Copyright 1996 Lawrence C. Marshhomoskedasticity assumption:
G = W
2 2
INVt = 1G + 1Dt + 2GVt + 2DtVt + 3GKt + 3DtKt + et
Dummy variable model assumes that :G = W
2 2
For Westinghouse observations Dt = 1; otherwise Dt = 0.
1W = 1G + 1
2W = 2G + 2
3W = 3G + 3
12.6
Copyright 1996 Lawrence C. Marsh
Problem with OLS on Each Equation
The first assumption of the Gauss-Markov Theorem concerns the model specification.
If the model is not fully and correctly specified the Gauss-Markov properties might not hold.
Any correlation of error terms across equations must be part of model specification.
12.7
Copyright 1996 Lawrence C. Marsh
Any correlation between the dependent variables of two or more equations that is not due to their explanatory variables is by default due to correlated error terms.
Correlated Error Terms12.8
Copyright 1996 Lawrence C. Marsh
1. Sales of Pepsi vs. sales of Coke.(uncontrolled factor: outdoor temperature)
2. Investments in bonds vs. investments in stocks. (uncontrolled factor: computer/appliance sales)
3. Movie admissions vs. Golf Course admissions. (uncontrolled factor: weather conditions)
4. Sales of butter vs. sales of bread. (uncontrolled factor: bagels and cream cheese)
Which of the following models wouldbe likely to produce positively correlatederrors and which would produce negatively correlations errors?
12.9
Copyright 1996 Lawrence C. Marsh
Joint Estimation of the Equations
INVGt = 1G + 2GVGt + 3GKGt + eGt
INVWt = 1W + 2WVWt + 3WKWt + eWt
cov(eGt, eWt) = GW
12.10
Copyright 1996 Lawrence C. Marsh
Seemingly Unrelated Regressions
When the error terms of two or more equationsare correlated, efficient estimation requires the useof a Seemingly Unrelated Regressions (SUR) type estimator to take the correlation into account.
Be sure to use the Seemingly Unrelated Regressions (SUR)procedure in your regression software program to estimateany equations that you believe might have correlated errors.
12.11
Copyright 1996 Lawrence C. Marsh
Separate vs. Joint Estimation
SUR will give exactly the same results as estimating each equation separately with OLS if either or both of the following two conditions are true:
1. Every equation has exactly the same set of explanatory variables with exactly the same values.
2. There is no correlation between the error terms of any of the equations.
12.12
Copyright 1996 Lawrence C. Marsh
Test for Correlation
GW
Test the null hypothesis of zero correlation
GW
G W
^
^ ^rGW 2
2 2
2 = T rGW
2
2(1)
asy.
12.13
Copyright 1996 Lawrence C. MarshStart withthe residuals eGt and eWt
from eachequation estimatedseparately.
^ ^
GW
G W
^
^ ^rGW 2
2 2
2
= T rGW
2 2(1)
asy.
GW eGteWt1T
^ ^^
G eGt1T
^^ 2 2
W eWt1T
^^ 2 2
12.14
Copyright 1996 Lawrence C. Marsh
Fixed Effects Model
yit = 1it + 2itx2it + 3itx3it + eit
yit = 1i + 2x2it + 3x3it + eit
Fixed effects models impose the restrictions:
1it = 1i 2it = 2 3it = 3
For each ith cross section in the tth time period:
Each ith cross-section has its own constant 1i intercept.
12.15
Copyright 1996 Lawrence C. Marsh
The Fixed Effects Model is conveniently represented using dummy variables:
yit = 11D1i + 12D2i + 13D3i + 14D4 i+ 2x2it + 3x3it + eit
D1i=1 if NorthD1i=0 if not N
D2i=1 if EastD2i=0 if not E
D3i=1 if SouthD3i=0 if not S
D4i=1 if WestD4i=0 if not W
yit = millions of bushels of corn producedx2it = price of corn in dollars per bushelx3it = price of soybeans in dollars per bushel
Each cross-sectional unit gets its own intercept,but each cross-sectional intercept is constant over time.
12.16
Copyright 1996 Lawrence C. Marsh
Ho : 11 = 12 = 13 = 14
Test for Equality of Fixed Effects
H1 : Ho not true
The Ho joint null hypothesis may be tested with F-statistic:
(SSER SSEU) / J
SSEU / (NT K)F = ~ F
(NT K)
J
SSER is the restricted error sum of squares (one intercept)SSEU is the unrestricted error sum of squares (four intercepts)N is the number of cross-sectional units (N = 4)K is the number of parameters in the model (K = 6)J is the number of restrictions being tested (J = N1 = 3)T is the number of time periods
12.17
Copyright 1996 Lawrence C. Marsh
Random Effects Model
yit = 1i + 2x2it + 3x3it + eit
1i = 1 + i
1 is the population mean intercept.
i is an unobservable random error thataccounts for the cross-sectional differences.
12.18
Copyright 1996 Lawrence C. Marsh
1i = 1 + i
i are independent of one another and of eit
E(i) = 0 var(i) = 2
where i = 1, ... ,N
Consequently, E(1i) = 1 var(1i) = 2
Random Intercept Term 12.19
Copyright 1996 Lawrence C. Marsh
yit = 1i + 2x2it + 3x3it + eit
yit = (1+i) + 2x2it + 3x3it + eit
yit = 1 + 2x2it + 3x3it + (i +eit)
yit = 1 + 2x2it + 3x3it + it
Random Effects Model 12.20
Copyright 1996 Lawrence C. Marsh
it = (i +eit)
yit = 1 + 2x2it + 3x3it + it
it has zero mean: E(it) = 0
it is homoskedastic: var(it) =+ e2 2
The errors from the same firm in different time periodsare correlated:
The errors from different firms are always uncorrelated:
cov(it,is) =2
cov(it,js) =
t ° s
i ° j
12.21
Copyright 1996 Lawrence C. Marsh
Simultaneous
Equations
Models
Chapter 13
Copyright © 1997 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein.
13.1
Copyright 1996 Lawrence C. Marsh
Keynesian Macro Model
Assumptions of Simple Keynesian Model
1. Consumption, c, is function of income, y.
2. Total expenditures = consumption + investment.
3. Investment assumed independent of income.
13.2
Copyright 1996 Lawrence C. Marsh
consumption is a function of income:
income is either consumed or invested:
c = 1 + 2 y
y = c + i
The Structural Equations 13.3
Copyright 1996 Lawrence C. Marsh
The Statistical Model
ct = 1 + 2 yt + et
yt = ct + it
The consumption equation:
The income identity:
13.4
Copyright 1996 Lawrence C. MarshThe Simultaneous Natureof Simultaneous Equations
ct = 1 + 2 yt + et
yt = ct + it
Since yt
contains et
they arecorrelated
2. 1.
3.
4.
5.
13.5
Copyright 1996 Lawrence C. Marsh
The Failure of Least Squares
The least squares estimators of parameters in a structural simul-taneous equation is biased andinconsistent because of the cor-relation between the random errorand the endogenous variables onthe right-hand side of the equation.
13.6
Copyright 1996 Lawrence C. Marsh
Single Equation: Simultaneous Equations:
Single vs. Simultaneous Equations
ct
yt
et
ct
yt i t
et
13.7
Copyright 1996 Lawrence C. Marsh
Deriving the Reduced Form
ct = 1 + 2 yt + et
yt = ct + it
ct = 1 + 2(ct + it) + et
(1 2)ct = 1 + 2 it + et
13.8
Copyright 1996 Lawrence C. MarshDeriving the Reduced Form
(1 2)ct = 1 + 2 it + et
ct = + it + et(12) (12) (12)11
2
ct = 11 + 21 it + t
The Reduced Form Equation
13.9
Copyright 1996 Lawrence C. Marsh
Reduced Form Equation
ct = 11 + 21 it + t
(12)111 = (12)
221 =
(12)1t = + et and
13.10
Copyright 1996 Lawrence C. Marsh
yt = ct + itwhere ct = 11 + 21 it + t
yt = 12 + 22 it + t
It is sometimes useful to give this equationits own reduced form parameters as follows:
yt = 11 + (1+21) it + t
13.11
Copyright 1996 Lawrence C. Marsh
yt = 12 + 22 it + t
ct = 11 + 21 it + t
Since ct and yt are related through the identity:
yt = ct + it , the error term, t, of these two
equations is the same, and it is easy to show that:
(12)111 = 12 =
(12)22 = (121) = 1
13.12
Copyright 1996 Lawrence C. Marsh
IdentificationThe structural parameters are 1 and 2.
The reduced form parameters are 11 and 21.
Once the reduced form parameters are estimated,the identification problem is to determine if theorginal structural parameters can be expresseduniquely in terms of the reduced form parameters.
(121)2 =
21^^
^(121)1 =
11^
^
^
13.13
Copyright 1996 Lawrence C. Marsh
Identification
An equation is exactly identified if its structural (behavorial) parameters can be uniquely expres-sed in terms of the reduced form parameters.
An equation is over-identified if there is morethan one solution for expressing its structural (behavorial) parameters in terms of the reducedform parameters.
An equation is under-identified if its structural (behavorial) parameters cannot be expressed in terms of the reduced form parameters.
13.14
Copyright 1996 Lawrence C. MarshThe Identification Problem
A system of M equations containing M endogenous variables must exclude at least M1 variables from a given equation in order for the parameters of that equation to be identified and to be able tobe consistently estimated.
13.15
Copyright 1996 Lawrence C. Marsh
Two Stage Least Squares
Problem: right-hand endogenous variablesyt2 and yt1 are correlated with the error terms.
yt1 = 1 + 2 yt2 + 3 xt1 + et1
yt2 = 1 + 2 yt1 + 3 xt2 + et2
13.16
Copyright 1996 Lawrence C. MarshProblem: right-hand endogenous variablesyt2 and yt1 are correlated with the error terms.
Solution: First, derive the reduced form equations.
yt1 = 1 + 2 yt2 + 3 xt1 + et1
yt2 = 1 + 2 yt1 + 3 xt2 + et2
yt1 = 11 + 21 xt1 + 31 xt2 + t1
yt2 = 12 + 22 xt1 + 32 xt2 + t2
Solve two equations for two unknowns, yt1, yt2 :
13.17
Copyright 1996 Lawrence C. Marsh
yt1 = 11 + 21 xt1 + 31 xt2 + t1
yt2 = 12 + 22 xt1 + 32 xt2 + t2
Use least squares to get fitted values:
2SLS: Stage I
yt1 = 11 + 21 xt1 + 31 xt2^ ^ ^ ^
yt2 = 12 + 22 xt1 + 32 xt2^ ^ ^ ^ yt2 = yt2 + t2
^ ^yt1 = yt1 + t1
^ ^
13.18
Copyright 1996 Lawrence C. Marsh2SLS: Stage II
yt2 = yt2 + t2^ ^yt1 = yt1 + t1
^ ^ and
yt1 = 1 + 2 yt2 + 3 xt1 + et1
yt2 = 1 + 2 yt1 + 3 xt2 + et2
Substitue in
for yt1 , yt2
yt1 = 1 + 2 (yt2 + t2) + 3 xt1 + et1^ ^
yt2 = 1 + 2 (yt1 + t1) + 3 xt2 + et2^^
13.19
Copyright 1996 Lawrence C. Marsh2SLS: Stage II (continued)
yt1 = 1 + 2 yt2 + 3 xt1 + ut1^
yt2 = 1 + 2 yt1 + 3 xt2 + ut2^
^ut1 = 2t2 + et1 ut2 = 2t1 + et2^where and
Run least squares on each of the above equationsto get 2SLS estimates:
1 , 2 , 3 , 1 , 2 and 3
~ ~ ~ ~ ~ ~
13.20
Copyright 1996 Lawrence C. Marsh
Nonlinear
Least
Squares
Chapter 14
Copyright © 1997 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein.
14.1
Copyright 1996 Lawrence C. Marsh
(A.) “Regression” model with only an intercept term:
Review of Least Squares Principle
yt = + et
et = yt
et = (yt )2 2
SSE = (yt )2
SSE = 2 (yt ) = 0^
yt = 0^
yt = 0^
= yt = y^ 1T
(minimize the sum of squared errors)
Yields an exact analytical solution:
14.2
Copyright 1996 Lawrence C. Marsh
Review of Least Squares(B.) Regression model without an intercept term:
yt = xt + et
et = yt xt
et = (yt xt)2 2
SSE = (yt xt)2
SSE = 2 xt(yt xt)= 0^
xtyt xt = 0^ 2
xt yt xt = 0^ 2
xt = xtyt^ 2
=^ xtyt
2xt
This yields an exact analytical solution:
14.3
Copyright 1996 Lawrence C. MarshReview of Least Squares(C.) Regression model with both an intercept and a slope:
yt = + xt + et SSE = (yt xt)2
SSE = 2 (yt xt) = 0^ ^
SSE = 2 xt(yt xt) = 0^ ^
y x = 0^ ^
=^ (xtx)(yty)
(xtx)2xtyt xt xt = 0^ ^ 2
This yields an exact analytical solution:
= y x^ ^
14.4
Copyright 1996 Lawrence C. MarshNonlinear Least Squares(D.) Nonlinear Regression model:
yt = xt + et
SSE = (yt xt)
2
PROBLEM: An exactanalytical solution tothis does not exist.
SSE = 2 xt
ln(xt)(yt xt) = 0
^ ^
xtln(xt)yt] xt
ln(xt)] = 0 ^^
Must use numericalsearch algorithm totry to find value of to satisfy this.
14.5
Copyright 1996 Lawrence C. Marsh
Find Minimum of Nonlinear SSE
^
SSE SSE = (yt xt)
2
14.6
Copyright 1996 Lawrence C. Marsh
The least squares principle is still appropriate when themodel is nonlinear, but it is harder to find the solution.
Conclusion14.7
Copyright 1996 Lawrence C. Marsh
Nonlinear least squares optimization methods:
The Gauss-Newton Method
Optional Appendix 14.8
Copyright 1996 Lawrence C. Marsh
The Gauss-Newton Algorithm
1. Apply the Taylor Series Expansion to the nonlinear model around some initial b(o).
2. Run Ordinary Least Squares (OLS) on the linear part of the Taylor Series to get b(m).
3. Perform a Taylor Series around the new b(m) to get b(m+1) .
4. Relabel b(m+1) as b(m) and rerun steps 2.-4.
5. Stop when (b(m+1) b(m) ) becomes very small.
14.9
Copyright 1996 Lawrence C. MarshThe Gauss-Newton Method
yt = f(Xt,b) + t for t = 1, . . . , n.
Do a Taylor Series Expansion around the vector b = b(o) as follows:
yt = f(Xt,b) + f’(Xt,b)(b - b) + t
where t؛ (b - b(o))
Tf’’(Xt,b)(b - b) + Rt + t
f(Xt,b) = f(Xt,b) + f’(Xt,b)(b - b)
+ (b - b)Tf’’(Xt,b)(b - b) + Rt
14.10
Copyright 1996 Lawrence C. Marshyt = f(Xt,b) + f’(Xt,b)(b - b) + t
yt - f(Xt,b) = f’(Xt,b)b - f’(Xt,b) b + t
yt - f(Xt,b) + f’(Xt,b) b = f’(Xt,b)b + t
yt = f’(Xt,b)b + t
where yt
؛yt - f(Xt,b) + f’(Xt,b) b
This is linear in b .
Gauss-Newton just runs OLS on thistransformed truncated Taylor series.
14.11
Copyright 1996 Lawrence C. Marsh
yt = f’(Xt,b)b + t
Gauss-Newton just runs OLS on thistransformed truncated Taylor series.
or y = f’(X,b)b + خ
for t = 1, . . . , n in matrix terms
b [ f’(X,b)T f’(X,b)]-1 f’(X,b)
T y
^
This is analogous to linear OLS where y = Xb + خ led to the solution: b XTX)XTy^except that X is replaced with the matrix of firstpartial derivatives: f’(Xt,b) and y is replaced by y
(i.e. “y” = y* and “X” = f’(X,b) )
14.12
Copyright 1996 Lawrence C. Marsh
Recall that: y*(o)
yf(X,b(o)) + f’(X,b) b؛
Now define: y
؛ y f(X,b(o))
Therefore: y
= y + f’(X,b) b
b [ f’(X,b)T f’(X,b)]-1 f’(X,b)
T y
^
Now substitute in for yin Gauss-Newton solution:
to get:
b = b(o) + [ f’(X,b)T f’(X,b)]-1 f’(X,b)
T y
^
14.13
Copyright 1996 Lawrence C. Marsh
b = b(o) + [ f’(X,b)T f’(X,b)]-1 f’(X,b)
T y
^
b(1) = b+ [ f’(X,b)T f’(X,b)]-1 f’(X,b)
T y
Now call this b value bas follows:^
More generally, in going from interation m toiteration (m+1) we obtain the general expression:
b(m+1) = b(m) + [ f’(X,b(m)T f’(X,b(m)]-1 f’(X,b(m)
T y(m)
14.14
Copyright 1996 Lawrence C. Marsh
b(m+1) = [ f’(X,b(m)T f’(X,b(m)]-1 f’(X,b(m))
T y*(m)
b(m+1) = b(m) + [ f’(X,b(m))T f’(X,b(m))]-1 f’(X,b(m))
T y(m)
Thus, the Gauss-Newton (nonlinear OLS) solutioncan be expressed in two alternative, but equivalent,forms:
1. replacement form:
2. updating form:
14.15
Copyright 1996 Lawrence C. Marsh
For example, consider Durbin’s Method of estimatingthe autocorrelation coefficient under a first-order autoregression regime:
y t = b1 + b2Xt 2 + . . . + bK Xt K + t for t = 1, . . . , n.
t = t - 1 + ut where u t satisfies the conditions
E u t = 0 , E u 2t = su2, E u t u s = 0 for s ° t.
Therefore, u t is nonautocorrelated and homoskedastic.
Durbin’s Method is to set aside a copy of the equation,lag it once, multiply by and subtract the new equation
from the original equation, then move the yt-1 term to
the right side and estimate along with the bs by OLS.
14.16
Copyright 1996 Lawrence C. Marsh
Durbin’s Method is to set aside a copy of the equation,lag it once, multiply by and subtract the new equation
from the original equation, then move the yt-1 term to
the right side and estimate along with the b’s by OLS.
y t = b1 + b2Xt 2 + b3 X t 3 + t for t = 1, . . . , n.
where t = t - 1 + ut
y t-1 = b1 + b2Xt -1, 2 + b3 Xt -1, 3 + t -1
Lag once and multiply by
Subtract from the original and move y t-1 to right side:
yt = b1-+ b2Xt 2 - Xt-1, 2 + b3(Xt 3 Xt-1, 3)+ y t-1+ ut
14.17
Copyright 1996 Lawrence C. Marsh
yt = b1-+ b2Xt 2 - Xt-1, 2 + b3(Xt 3 -Xt-1, 3) + y t-1+ ut
Now Durbin separates out the terms as follows:
yt = b1-+ b2Xt 2 - b2Xt-1 2 + b3Xt 3 - b3Xt-1 3+ y t-1+ ut
The structural (restricted,behavorial) equation is:
The corresponding reduced form (unrestricted) equation is:
yt = 1+ 2Xt, 2 + 3Xt-1, 2 + 4Xt, 3 + 5Xt-1, 3 + 6yt-1+ u t
1 = b1- 2 = b23= - b2 4 = b3 5= - b3 6=
14.18
Copyright 1996 Lawrence C. Marsh
Given OLS estimates: 1 2 3 4 5 6
^^^^^ ^
we can get three separate and distinct estimates for
3
2
^ ^
^
5
4^
^^ 6^^
These three separate estimates of are in conflict !!!It is difficult to know which one to use as “the”legitimate estimate of Durbin used the last one.
1 = b1- 2 = b23= - b2 4 = b3 5= - b3 6=
14.19
Copyright 1996 Lawrence C. Marsh
The problem with Durbin’s Method is that it ignoresthe inherent nonlinear restrictions implied by this structural model. To get a single (i.e. unique) estimatefor the implied nonlinear restrictions must be incorporated directly into the estimation process.
Consequently, the above structural equation should beestimated using a nonlinear method such as theGauss-Newton algorithm for nonlinear least squares.
yt = b1-+ b2Xt 2 - b2Xt -1, 2 + b3Xt 3 - b3Xt -1, 3+ yt-1+ ut
14.20
Copyright 1996 Lawrence C. Marsh
yt = b1-+ b2Xt 2 - b2Xt-1, 2 + b3Xt 3 - b3Xt-1, 3+ yt-1+ ut
f’(Xt,b) [ ]yt
= =X t, 2 X t-1,2)
=X t, 3 X t-1,3)
yt
= ( - b1-b2Xt-1,2 - b3Xt-1,3+ y t-1 )
yt
b1
yt
b2
yt
b3
yt
b1
yt
b2
yt
b3
14.21
Copyright 1996 Lawrence C. Marsh
where yt(m)
yt - f(Xt,bm) + f’(Xt,b(m) b(m؛
(m+1) = [ f’(X,bm)T f’(X,bm)]-1 f’(X,b(m)
T y(m
^
f(Xt,b) = b1-+ b2Xt 2 - b2Xt-1 2 + b3Xt 3 - b3Xt-1 3+ y t-1
b(m) =
b
1(m)
(m)
b2(m)
b3(m)
Iterate until convergence.
f’(Xt,bm [ ]yt
(m)
yt
b1(m)
yt
b2(m)
yt
b3(m)
14.22
Copyright 1996 Lawrence C. Marsh
Distributed
Lag Models
Chapter 15
Copyright © 1997 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein.
15.1
Copyright 1996 Lawrence C. Marsh
The Distributed Lag Effect
Economic actionat time t
Effect at time t
Effect at time t+1
Effect at time t+2
15.2
Copyright 1996 Lawrence C. Marsh
Unstructured Lags
yt = + 0 xt + 1 xt-1 + 2 xt-2 + . . . +n xt-n + et
“n” unstructured lags
no systematic structure imposed on the ’s
the ’s are unrestricted
15.3
Copyright 1996 Lawrence C. Marsh
Problems with Unstructured Lags
1. n observations are lost with n-lag setup.
2. high degree of multicollinearity among xt-j’s.
3. many degrees of freedom used for large n.
4. could get greater precision using structure.
15.4
Copyright 1996 Lawrence C. Marsh
The Arithmetic Lag Structure
proposed by Irving Fisher (1937)
the lag weights decline linearly
Imposing the relationship:
# = (n - # + 1)
0 = (n+1) 1 = n 2 = (n-1) 3 = (n-2) n-2 = 3 n-1 = 2 n = only need to estimate one coefficient, ,
instead of n+1 coefficients, 0 , ... , n .
15.5
Copyright 1996 Lawrence C. Marsh
Arithmetic Lag Structure
yt = + 0 xt + 1 xt-1 + 2 xt-2 + . . . +n xt-n + et
yt = + (n+1) xt + n xt-1 + (n-1) xt-2 + . . . + xt-n + et
Step 1: impose the restriction: # = (n - # + 1)
Step 2: factor out the unknown coefficient, .
yt = + [(n+1)xt + nxt-1 + (n-1)xt-2 + . . . + xt-n] + et
15.6
Copyright 1996 Lawrence C. Marsh
Arithmetic Lag Structure
Step 3: Define zt .
yt = + [(n+1)xt + nxt-1 + (n-1)xt-2 + . . . + xt-n] + et
zt = [(n+1)xt + nxt-1 + (n-1)xt-2 + . . . + xt-n]
Step 5: Run least squares regression on:
yt = + zt + et
Step 4: Decide number of lags, n.
For n = 4: zt = [ 5xt + 4xt-1 + 3xt-2 + 2xt-3 + xt-4]
15.7
Copyright 1996 Lawrence C. Marsh
Arithmetic Lag Structure
i
i
0 = (n+1)
1 = n
2 = (n-1)
n =
.
.
.
0 1 2 . . . . . n n+1
..
.
.
linear lag structure
15.8
Copyright 1996 Lawrence C. Marsh
Polynomial Lag Structureproposed by Shirley Almon (1965)
the lag weights fit a polynomial
where i = 1, . . . , n p = 2 and n = 4
For example, a quadratic polynomial:0 = 0
1 = 0 + 1 + 2
2 = 0 + 21 + 42
3 = 0 + 31 + 92
4 = 0 + 41 + 162
n = the length of the lagp = degree of polynomial
where i = 1, . . . , ni = 0 + 1i + 2i +...+ pi 2 p
i = 0 + 1i + 2i 2
15.9
Copyright 1996 Lawrence C. Marsh
Polynomial Lag Structureyt = + 0 xt + 1 xt-1 + 2 xt-2 + 3 xt-3 +4 xt-4 + et
yt = + 0xt + 0 + 1 + 2xt-1 + (0 + 21 + 42)xt-2
+ (0 + 31 + 92)xt-3+ (0 + 41 + 162)xt-4 + et
Step 2: factor out the unknown coefficients: 0, 1, 2.
yt = + 0 [xt + xt-1 + xt-2 + xt-3 + xt-4] + 1 [xt + xt-1 + 2xt-2 + 3xt-3 + 4xt-4] + 2 [xt + xt-1 + 4xt-2 + 9xt-3 + 16xt-4] + et
Step 1: impose the restriction: i = 0 + 1i + 2i 2
15.10
Copyright 1996 Lawrence C. Marsh
Polynomial Lag Structure
Step 3: Define zt0 , zt1 and zt2 for 0 , 1 , and 2.
yt = + 0 [xt + xt-1 + xt-2 + xt-3 + xt-4] + 1 [xt + xt-1 + 2xt-2 + 3xt-3 + 4xt-4] + 2 [xt + xt-1 + 4xt-2 + 9xt-3 + 16xt-4] + et
z t0 = [xt + xt-1 + xt-2 + xt-3 + xt-4]
z t1 = [xt + xt-1 + 2xt-2 + 3xt-3 + 4xt- 4 ]
z t2 = [xt + xt-1 + 4xt-2 + 9xt-3 + 16xt- 4]
15.11
Copyright 1996 Lawrence C. Marsh
Polynomial Lag Structure
Step 4: Regress yt on zt0 , zt1 and zt2 .
yt = + 0 z t0 + 1 z t1 + 2 z t2 + et
Step 5: Express i‘s in terms of 0 , 1 , and 2.
^ ^ ^ ^
0 = 0
1 = 0 + 1 + 2
2 = 0 + 21 + 42
3 = 0 + 31 + 92
4 = 0 + 41 + 162
^^
^
^
^
^
^ ^ ^
^ ^ ^
^ ^ ^
^ ^ ^
15.12
Copyright 1996 Lawrence C. Marsh
Polynomial Lag Structure
.. . .
.
0 1 2 3 4 i
i
Figure 15.3
0
1
23
4
15.13
Copyright 1996 Lawrence C. Marsh
Geometric Lag Structure
yt = + 0 xt + 1 xt-1 + 2 xt-2 + . . . + et
infinite distributed lag model:
yt = + i xt-i + eti = 0
(15.3.1)
geometric lag structure:
i = i where || < 1 and i
15.14
Copyright 1996 Lawrence C. Marsh
Geometric Lag Structure
yt = + 0 xt + 1 xt-1 + 2 xt-2 + 3 xt-3 + . . . + et
yt = + xt + xt-1 + xt-2 + xt-3 + . . .) + et
infinite unstructured lag:
infinite geometric lag:
Substitute i = i
0 = 1 = 2 = 2
3 = 3
. ..
15.15
Copyright 1996 Lawrence C. Marsh
Geometric Lag Structure
interim multiplier (3-period) :
impact multiplier :
long-run multiplier :
yt = + xt + xt-1 + xt-2 + xt-3 + . . .) + et
+ +
+ + + + . . .
15.16
Copyright 1996 Lawrence C. Marsh
Geometric Lag Structure
i
Figure 15.5
.
..
. .0 1 2 3 4 i
1 =
2 = 2
3 = 3
4 = 4
0 =
geometrically declining weights
15.17
Copyright 1996 Lawrence C. Marsh
Geometric Lag Structure
Problem:
How to estimate the infinite number of geometric lag coefficients ???
yt = + xt + xt-1 + xt-2 + xt-3 + . . .) + et
Answer:
Use the Koyck transformation.
15.18
Copyright 1996 Lawrence C. Marsh
The Koyck Transformation
yt = + xt + xt-1 + xt-2 + xt-3 + . . .) + et
yt yt-1 = + xt + (et et-1)
Lag everything once, multiply by and subtract from original:
yt-1 = + xt-1 + xt-2 + xt-3 + . . .) + et-1
15.19
Copyright 1996 Lawrence C. Marsh
The Koyck Transformation
yt yt-1 = + xt + (et et-1)
yt = + yt-1 + xt + (et et-1)
Solve for yt by adding yt-1 to both sides:
yt = + yt-1 + xt + t
15.20
Copyright 1996 Lawrence C. Marsh
The Koyck Transformationyt = + yt-1 + xt + (et et-1)
yt = + yt-1 + xt + t
Defining = , = , and = ,
use ordinary least squares:
= ^ ^
= ^ ^
= ^ ^ ^
The original structuralparameters can now beestimated in terms ofthese reduced formparameter estimates.
15.21
Copyright 1996 Lawrence C. Marsh
Geometric Lag Structure
0 =
1 =
2 = 2
3 = 3
. ..
^ ^
^ ^ ^
^ ^ ^
^ ^ ^
yt = + xt + xt-1 + xt-2 + xt-3 + . . .) + et^ ^ ^ ^ ^ ^
yt = + 0 xt + 1 xt-1 + 2 xt-2 + 3 xt-3 + . . . + et ^^ ^ ^ ^ ^
15.22
Copyright 1996 Lawrence C. Marsh
Durbin’s h-test for autocorrelation
T 11 ( T 1)[se(b2)]2
h = 1 d2
h = Durbin’s h-test statistic
d = Durbin-Watson test statistic
se(b2) = standard error of the estimate b2
T = sample size
Estimates inconsistent if geometric lag model is autocorrelated,but Durbin-Watson test is biased in favor of no autocorrelation.
15.23
Copyright 1996 Lawrence C. Marsh
yt = + x*t + et
Adaptive Expectations
yt = credit card debt
x*t = expected (anticipated) income
(x*t is not observable)
15.24
Copyright 1996 Lawrence C. MarshAdaptive Expectations
x*t - x*t-1 = (xt-1 - x*t-1)
adjust expectations based on past realization:
15.25
Copyright 1996 Lawrence C. MarshAdaptive Expectations
x*t - x*t-1 = (xt-1 - x*t-1)
x*t = xt-1 + (1- ) x*t-1
rearrange to get:
xt-1 = [x*t - (1- ) x*t-1]
or
15.26
Copyright 1996 Lawrence C. MarshAdaptive Expectations
yt = + x*t + et
Lag this model once and multiply by (1):
yt = - (1)yt-1+ [x*t - (1)x*t-1]
+ et - (1)et-1
subtract this from the original to get:
(1)yt-1 = (1) + (1) x*t-1 + (1)et-1
15.27
Copyright 1996 Lawrence C. MarshAdaptive Expectations
yt = - (1)yt-1+ [x*t - (1)x*t-1]
+ et - (1)et-1
Since xt-1 = [x*t - (1- ) x*t-1] we get:
yt = - (1)yt-1+ xt-1 + ut
where ut = et - (1)et-1
15.28
Copyright 1996 Lawrence C. MarshAdaptive Expectations
yt = - (1)yt-1+ xt-1 + ut
yt = 1 + 2yt-1+ 3xt-1 + ut
Use ordinary least squares regression on:
and we get:
=(12)
3
^ ^
^ = (12)
^ ^ =(12)
1
^ ^
^
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Copyright 1996 Lawrence C. Marsh
Partial Adjustment
yt - yt-1 = (y*t - yt-1)
inventories partially adjust , 0 < < 1,
towards optimal or desired level, y*t :
y*t = + xt + et
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Copyright 1996 Lawrence C. MarshPartial Adjustment
yt - yt-1 = (y*t - yt-1)
= ( + xt + et - yt-1)
= + xt - yt-1+ et
yt = + (1 - yt-1 + xt + et
Solving for yt :
15.31
Copyright 1996 Lawrence C. MarshPartial Adjustment
yt = + (1 - yt-1 + xt + et
yt = 1 + 2yt-1+ 3xt + t
=(12)
3
^^
^ = (12) ^ ^
=(12)
1
^^
^
Use ordinary least squares regression to get:
15.32
Copyright 1996 Lawrence C. Marsh
Time Series
Analysis
Chapter 16
Copyright © 1997 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein.
16.1
Copyright 1996 Lawrence C. Marsh
Previous Chapters used Economic Models
1. economic model for dependent variable of interest.
2. statistical model consistent with the data.
3. estimation procedure for parameters using the data.
4. forecast variable of interest using estimated model.
Times Series Analysis does not use this approach.
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Copyright 1996 Lawrence C. Marsh
Time Series Analysis is useful for short term forecasting only.
Time Series Analysis does not generallyincorporate all of the economic relationships found in economic models.
Times Series Analysis uses more statistics and less economics.
Long term forecasting requires incorporating more involvedbehavioral economic relationships into the analysis.
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Copyright 1996 Lawrence C. Marsh
Univariate Time Series Analysis can be used to relate the current values of a single economicvariable to:
1. its past values
2. the values of current and past random errors
Other variables are not used in univariate time series analysis.
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Copyright 1996 Lawrence C. Marsh
1. autoregressive (AR)
2. moving average (MA)
3. autoregressive moving average (ARMA)
Three types of Univariate Time Series Analysisprocesses will be discussed in this chapter:
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Copyright 1996 Lawrence C. Marsh
1. its past values.
2. the past values of the other forecasted variables.
3. the values of current and past random errors.
Multivariate Time Series Analysis can be used to relate the current value of each of several economic variables to:
Vector autoregressive models discussed later inthis chapter are multivariate time series models.
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Copyright 1996 Lawrence C. Marsh
First-Order Autoregressive Processes, AR(1):
yt = + 1yt-1+ et, t = 1, 2,...,T. (16.1.1)
is the intercept.
1 is parameter generally between -1 and +1.
et is an uncorrelated random error with mean zero and variance e
.
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Copyright 1996 Lawrence C. Marsh
Autoregressive Process of order p, AR(p) :
yt = + 1yt-1 + 2yt-2 +...+ pyt-p + et (16.1.2)
is the intercept.
i’s are parameters generally between -1 and +1.
et is an uncorrelated random error with mean zero and variance e
.
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Copyright 1996 Lawrence C. Marsh
AR models always have one or more lagged dependent variables on the right hand side.
Consequently, least squares is no longer abest linear unbiased estimator (BLUE), but it does have some good asymptotic properties including consistency.
Properties of least squares estimator:
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Copyright 1996 Lawrence C. Marsh
AR(2) model of U.S. unemployment rates
yt = 0.5051 + 1.5537 yt-1 - 0.6515 yt-2
(0.1267) (0.0707) (0.0708)
Note: Q1-1948 through Q1-1978 from J.D.Cryer (1986) see unempl.dat
positive
negative
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Copyright 1996 Lawrence C. Marsh
Choosing the lag length, p, for AR(p):
The Partial Autocorrelation Function (PAF)
The PAF is the sequence of correlations between(yt and yt-1), (yt and yt-2), (yt and yt-3), and so on,given that the effects of earlier lags on yt are held constant.
16.11
Copyright 1996 Lawrence C. Marsh Partial Autocorrelation Function
yt = 0.5 yt-1 + 0.3 yt-2 + et
02 / T
2 / T
1
1
k
kk is the last (kth) coefficient in a kth order AR process.
This sample PAF suggests a second order process AR(2) which is correct.
Data simulatedfrom this model:
kk^
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Copyright 1996 Lawrence C. Marsh
Using AR Model for Forecasting:
unemployment rate: yT-1 = 6.63 and yT = 6.20
yT+1 = + 1 yT + 2 yT-1
= 0.5051 + (1.5537)(6.2) - (0.6515)(6.63)
= 5.8186
^ ^ ^ ^
yT+2 = + 1 yT+1 + 2 yT
= 0.5051 + (1.5537)(5.8186) - (0.6515)(6.2)
= 5.5062
^ ^ ^ ^
yT+1 = + 1 yT + 2 yT-1
= 0.5051 + (1.5537)(5.5062) - (0.6515)(5.8186)
= 5.2693
^ ^ ^ ^
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Copyright 1996 Lawrence C. Marsh
Moving Average Process of order q, MA(q):
yt = + et + 1et-1 + 2et-2 +...+ qet-q + et (16.2.1)
is the intercept.
i‘s are unknown parameters.
et is an uncorrelated random error with mean zero and variance e
.
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Copyright 1996 Lawrence C. Marsh
An MA(1) process:
yt = + et + 1et-1 (16.2.2)
Minimize sum of least squares deviations:
S(,1) = et = yt - -1et-1) (16.2.3)2
t=1
T
t=1
T 2
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Copyright 1996 Lawrence C. Marsh
stationary:A stationary time series is one whose mean, variance,and autocorrelation function do not change over time.
nonstationary:A nonstationary time series is one whose mean,variance or autocorrelation function change over time.
Stationary vs. Nonstationary 16.16
Copyright 1996 Lawrence C. Marsh
yt = z t - z t-1
First Differencing is often used to transforma nonstationary series into a stationary series:
where z t is the original nonstationary series
and yt is the new stationary series.
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Copyright 1996 Lawrence C. Marsh
Choosing the lag length, q, for MA(q):
The Autocorrelation Function (AF)
The AF is the sequence of correlations between(yt and yt-1), (yt and yt-2), (yt and yt-3), and so on,without holding the effects of earlier lags on yt constant.
The PAF controlled for the effects of previous lagsbut the AF does not control for such effects.
16.18
Copyright 1996 Lawrence C. MarshAutocorrelation Function
yt = et 0.9 et-1
02 / T
2 / T
1
1
k
rkk
rkk is the last (kth) coefficient in a kth order MA process.
This sample AF suggests a first order process MA(1) which is correct.
Data simulatedfrom this model:
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Copyright 1996 Lawrence C. Marsh
Autoregressive Moving AverageARMA(p,q)
An ARMA(1,2) has one autoregressive lagand two moving average lags:
yt = + 1yt-1 + et + 1et-1 + 2 et-2
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Copyright 1996 Lawrence C. Marsh
Integrated Processes
A time series with an upward or downwardtrend over time is nonstationary.
Many nonstationary time series can be made stationary by differencing them one or more times.
Such time series are called integrated processes.
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Copyright 1996 Lawrence C. Marsh
The number of times a series must be differenced to make it stationary is theorder of the integrated process, d.
An autocorrelation function, AF, with large, significant autocorrelationsfor many lags may require more thanone differencing to become stationary.
Check the new AF after each differencingto determine if further differencing is needed.
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Copyright 1996 Lawrence C. Marsh
Unit Root
zt = 1zt -1 + + et + 1et -1 (16.3.2)
-1 < 1 < 1 stationary ARMA(1,1)
1 = 1 nonstationary process
1 = 1 is called a unit root
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Copyright 1996 Lawrence C. Marsh
Unit Root Tests
zt = 1zt -1 + + et + 1et -1 (16.3.3)
Testing1 = 0 is equivalent to testing 1 = 1
zt - zt -1 = (1- 1)zt -1 + + et + 1et -1
*
where zt = zt - zt -1 and 1 = 1- 1 *
*
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Copyright 1996 Lawrence C. Marsh
Unit Root Tests
H0:1 = 0 vs. H1:1 < 0 (16.3.4)**
Computer programs typically use one of the following tests for unit roots:
Dickey-Fuller Test
Phillips-Perron Test
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Copyright 1996 Lawrence C. Marsh
Autoregressive Integrated Moving Average ARIMA(p,d,q)
An ARIMA(p,d,q) model represents an AR(p) - MA(q) process that has been differenced (integrated, I(d)) d times.
yt = + 1yt-1 +...+ pyt-p + et + 1et-1 +... + q et-q
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Copyright 1996 Lawrence C. Marsh
The Box-Jenkins approach:
1. Identificationdetermining the values of p, d, and q.
2. Estimationlinear or nonlinear least squares.
3. Diagnostic Checkingmodel fits well with no autocorrelation?
4. Forecastingshort-term forecasts of future yt values.
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Copyright 1996 Lawrence C. Marsh
Vector Autoregressive (VAR) Models
yt = 0+ 1yt-1 +...+ pyt-p + 1xt-1 +... + p xt-p + et
xt = 0+ 1yt-1 +...+ pyt-p + 1xt-1 +... + p xt-p + ut
Use VAR for two or more interrelated time series:
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Copyright 1996 Lawrence C. Marsh
1. extension of AR model.
2. all variables endogenous.
3. no structural (behavioral) economic model.
4. all variables jointly determined (over time).
5. no simultaneous equations (same time).
Vector Autoregressive (VAR) Models16.29
Copyright 1996 Lawrence C. Marsh
The random error terms in a VAR modelmay be correlated if they are affected byrelevant factors that are not in the modelsuch as government actions or national/international events, etc.
Since VAR equations all have exactly the same set of explanatory variables, the usualseemingly unrelation regression estimationproduces exactly the same estimates asleast squares on each equation separately.
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Copyright 1996 Lawrence C. Marsh
Consequently, regardless of whetherthe VAR random error terms arecorrelated or not, least squares estimationof each equation separately will provideconsistent regression coefficient estimates.
Least Squares is Consistent
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Copyright 1996 Lawrence C. Marsh
VAR Model Specification
To determine length of the lag, p, use:
2. Schwarz’s SIC criterion
1. Akaike’s AIC criterion
These methods were discussed in Chapter 15.
16.32
Copyright 1996 Lawrence C. Marsh
Spurious Regressions
yt = 1+ 2 xt + t
where t = 1 t-1 + t
-1 <1 < 1 I(0) (i.e. d=0)
1 = 1 I(1) (i.e. d=1)
If 1 =1 least squares estimates of 2 mayappear highly significant even when true 2 = 0 .
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Copyright 1996 Lawrence C. Marsh
Cointegration
yt = 1+ 2 xt + t
If xt and yt are nonstationary I(1)
we might expect that t is also I(1).
However, if xt and yt are nonstationary I(1)
but t is stationary I(0), then xt and yt are
said to be cointegrated.
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Copyright 1996 Lawrence C. Marsh
Cointegrated VAR(1) Model
yt = 0+ 1yt-1 + 1xt-1 + et
xt = 0+ 1yt-1 + 1xt-1 + ut
VAR(1) model:
If xt and yt are both I(1) and are cointegrated,
use an Error Correction Model, instead of VAR(1).
16.35
Copyright 1996 Lawrence C. Marsh
Error Correction Model
yt = 0+ (1-1)yt-1 + 1xt-1 + et
xt = 0+ 1yt-1 + (1-1)xt-1 + ut
yt = yt - yt-1 and xt = xt - xt-1
(continued)
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Copyright 1996 Lawrence C. Marsh
Error Correction Model
yt = 0+ 1(yt-1 - 1- 2 xt-1) + et *
xt = 0+ 2(yt-1 - 1- 2 xt-1) + ut *
0= 0 + 11
*
0= 0 + 21* 2= 1
1=1 1
1 - 12=1
1- 1
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Copyright 1996 Lawrence C. Marsh
yt-1 = 1+ 2 xt-1 + t-1
Estimate by least squares:
to get the residuals:
t-1 = yt-1 - 1- 2 xt-1 ^ ^ ^
Estimating an Error Correction Model
Step 1:Step 1:
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Copyright 1996 Lawrence C. Marsh
Estimate by least squares:
Estimating an Error Correction Model
Step 2:Step 2:
yt = 0+ 1 t-1 + et *
xt = 0+ 2 t-1 + ut *
^
^
16.39
Copyright 1996 Lawrence C. Marsh
Using cointegrated I(1) variables in aVAR model expressed solely in termsof first differences and lags of firstdifferences is a misspecification.
The correct specification is to use an
Error Correction Model
16.40
Copyright 1996 Lawrence C. Marsh
Chapter 17
Copyright © 1997 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein.
Guidelines forResearch Project
17.1
Copyright 1996 Lawrence C. Marsh
����
Formulation economic ====> econometric.
Estimation selecting appropriate method.
Interpretation how the xt’s impact on the yt .
Inference testing, intervals, prediction.
What Book Has Covered17.2
Copyright 1996 Lawrence C. Marsh
Topics for This Chapter1. Types of Data by Source
2. Nonexperimental Data
3. Text Data vs. Electronic Data
4. Selecting a Topic
5. Writing an Abstract
6. Research Report Format
17.3
Copyright 1996 Lawrence C. Marsh
Types of Data by Source
i) Experimental Data from controlled experiments.
ii) Observational Datapassively generated by society.
iii) Survey Datadata collected through interviews.
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Copyright 1996 Lawrence C. Marsh
Time vs. Cross-Section
Time Series Datadata collected at distinct points in time(e.g. weekly sales, daily stock price, annual budget deficit, monthly unemployment.)
Cross Section Datadata collected over samples of units, individuals, households, firms at a particular point in time. (e.g. salary, race, gender, unemployment by state.)
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Copyright 1996 Lawrence C. Marsh
Micro vs. Macro
Micro Data:data collected on individual economicdecision making units such as individuals,households or firms.
Macro Data:data resulting from a pooling or aggregatingover individuals, households or firms at thelocal, state or national levels.
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Copyright 1996 Lawrence C. Marsh
Flow vs. Stock
Flow Data:outcome measured over a period of time,such as the consumption of gasoline duringthe last quarter of 1997.
Stock Data:outcome measured at a particular point intime, such as crude oil held by Chevron inUS storage tanks on April 1, 1997.
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Copyright 1996 Lawrence C. Marsh
Quantitative vs. Qualitative
Quantitative Data:outcomes such as prices or income that maybe expressed as numbers or some transfor-mation of them (e.g. wages, trade deficit).
Qualitative Data:outcomes that are of an “either-or” nature(e.g. male, home owner, Methodist, bought car last year, voted in last election).
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Copyright 1996 Lawrence C. Marsh
International Data
International Financial Statistics (IMF monthly).
Basic Statistics of the Community (OECD annual).
Consumer Price Indices in the European Community (OECD annual).
World Statistics (UN annual).
Yearbook of National Accounts Statistics (UN).
FAO Trade Yearbook (annual).
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Copyright 1996 Lawrence C. Marsh
United States Data
Survey of Current Business (BEA monthly).
Handbook of Basic Economic Statistics (BES).
Monthly Labor Review (BLS monthly).
Federal Researve Bulletin (FRB monthly).
Statistical Abstract of the US (BC annual).
Economic Report of the President (CEA annual).
Economic Indicators (CEA monthly).
Agricultural Statistics (USDA annual).
Agricultural Situation Reports (USDA monthly).
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Copyright 1996 Lawrence C. Marsh
State and Local Data
State and Metropolitan Area Data Book (Commerce and BC, annual).
CPI Detailed Report (BLS, annual).
Census of Population and Housing (Commerce, BC, annual).
County and City Data Book (Commerce, BC, annual).
17.11
Copyright 1996 Lawrence C. Marsh
Citibase on CD-ROM
• Financial series: interest rates, stock market, etc.
• Business formation, investment and consumers.
• Construction of housing.
• Manufacturing, business cycles, foreign trade.
• Prices: producer and consumer price indexes.
• Industrial production.
• Capacity and productivity.
• Population.
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Copyright 1996 Lawrence C. Marsh
Citibase on CD-ROM(continued)
• Labor statistics: unemployment, households.
• National income and product accounts in detail.
• Forecasts and projections.
• Business cycle indicators.
• Energy consumption, petroleum production, etc.
• International data series including trade statistics.
17.13
Copyright 1996 Lawrence C. Marsh
Resources for Economists
Resources for Economists by Bill Goffe
http://econwpa.wustl.edu/EconFAQ/EconFAQ.html
Bill Goffe provides a vast database of information about the economics profession including economic organizations, working papers and reports, and economic data series.
17.14
Copyright 1996 Lawrence C. Marsh
Internet Data Sources
• Shortcut to All Resources.
• Macro and Regional Data.
• Other U.S. Data.
• World and Non-U.S. Data.
• Finance and Financial Markets.
• Data Archives.
• Journal Data and Program Archives.
A few of the items on Bill Goffe’s Table of Contents:
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Copyright 1996 Lawrence C. Marsh
Useful Internet Addresses
http://seamonkey.ed.asu.edu/~behrens/teach/WWW_data.html
http://www.sims.berkeley.edu/~hal/pages/interesting.html
http://www.stls.frb.org FED RESERVE BK - ST. LOUIS
http://www.bls.gov BUREAU OF LABOR STATISTICS
http://nber.harvard.edu NAT’L BUR. ECON. RESEARCH
http://www.inform.umd.edu:8080/EdRes/Topic/EconData/.www/econdata.html UNIVERSITY OF MARYLAND
http://www.bog.frb.fed.us FEB BOARD OF GOVERNORS
http://www.webcom.com/~yardeni/economic.html
17.16
Copyright 1996 Lawrence C. MarshData from Surveys
i) identify the population of interest.
ii) designing and selecting the sample.
iii) collecting the information.
iv) data reduction, estimation and inference.
The survey process has four distinct aspects:
17.17
Copyright 1996 Lawrence C. Marsh
Controlled Experiments
1. Labor force participation: negative income tax:guaranteed minimum income experiment.
2. National cash housing allowance experiment:impact on demand and supply of housing.
3. Health insurance: medical cost reduction: sensitivity of income groups to price change.
4. Peak-load pricing and electricity use:daily use pattern of residential customers.
Controlled experiments were done on these topics:
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Copyright 1996 Lawrence C. Marsh
Economic Data Problems
I. poor implicit experimental design
(i) collinear explanatory variables.
(ii) measurement errors.
II. inconsistent with theory specification
(i) wrong level of aggregation.
(ii) missing observations or variables.
(iii) unobserved heterogeneity.
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Copyright 1996 Lawrence C. Marsh
Selecting a Topic
• “What am I interested in?”
• Well-defined, relatively simple topic.
• Ask prof for ideas and references.
• Journal of Economic Literature (ECONLIT)
• Make sure appropriate data are available.
• Avoid extremely difficult econometrics.
• Plan your work and work your plan.
General tips for selecting a research topic:
�������
17.20
Copyright 1996 Lawrence C. Marsh
Writing an Abstract
(i) concise statement of the problem.
(ii) key references to available information.
(iii) description of research design including:(a) economic model(b) statistical model(c) data sources(d) estimation, testing and prediction
(iv) contribution of the work
Abstract of less than 500 words should include:
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Copyright 1996 Lawrence C. Marsh
Research Report Format
1. Statement of the Problem. 2. Review of the Literature. 3. The Economic Model. 4. The Statistical Model. 5. The Data. 6. Estimation and Inferences Procedures. 7. Empirical Results and Conclusions. 8. Possible Extensions and Limitations. 9. Acknowledgments.10. References.
17.22