the simple regression model
Post on 07-Jan-2016
31 Views
Preview:
DESCRIPTION
TRANSCRIPT
FIN357 Li 1
The Simple Regression Model
y = 0 + 1x + u
FIN357 Li 2
Some Terminology
In the simple linear regression model, where y = 0 + 1x + u, we typically refer to y as the Dependent Variable, or Left-Hand Side Variable, or Explained Variable, or
FIN357 Li 3
Some Terminology
we typically refer to x as the Independent Variable, or Right-Hand Side Variable, or Explanatory Variable, or Regressor, or Control Variables
FIN357 Li 4
A Simple Assumption
The average value of u, the error term, in the population is 0. That is,
E(u) = 0
FIN357 Li 5
We also assume
E(u|x) = 0
E(y|x) = 0 + 1x
FIN357 Li 6
..
x1 x2
E(y|x) as a linear function of x, where for any x the distribution of y is centered about E(y|x)
E(y|x) = 0 + 1x
y
f(y)
FIN357 Li 7
Ordinary Least Squares (OLS)
Let {(xi,yi): i=1, …,n} denote a random sample of size n from the population
For each observation in this sample, it will be the case that
yi = 0 + 1xi + ui
FIN357 Li 8
.
..
.
y4
y1
y2
y3
x1 x2 x3 x4
}
}
{
{
u1
u2
u3
u4
x
y
Population regression line, sample data pointsand the associated error terms
E(y|x) = 0 + 1x
FIN357 Li 9
Basic idea of regression is to estimate the population parameters from a sample
Intuitively, OLS is fitting a line through the sample points such that the sum of squared residuals is as small as possible.
The residual, û, is an estimate of the error term, u, and is the difference between the fitted line (sample regression function) and the sample point
FIN357 Li 10
.
..
.
y4
y1
y2
y3
x1 x2 x3 x4
}
}
{
{
û1
û2
û3
û4
x
y
Sample regression line, sample data pointsand the associated estimated error terms (residuals)
xy 10ˆˆˆ
FIN357 Li 11
One approach to estimate coefficients
Given the intuitive idea of fitting a line, we can set up a formal minimization problem
That is, we want to choose our parameters such that we minimize the following:
n
iii
n
ii xyu
1
2
101
2 ˆˆˆ
FIN357 Li 12
It could be shown that estimated coefficient is
1
12
1
ˆ
n
i ii
n
ii
x x y y
x x
FIN357 Li 13
Summary of OLS slope estimate
The slope estimate is the sample covariance between x and y divided by the sample variance of x
If x and y are positively correlated, the slope will be positive
If x and y are negatively correlated, the slope will be negative
FIN357 Li 14
Algebraic Properties of OLS: in English
The sum of the OLS residuals is zero
Thus, the sample average of the OLS residuals is zero as well
The sample covariance between the regressors and the OLS residuals is zero
The OLS regression line always goes through the mean of the sample
FIN357 Li 15
Algebraic Properties of OLS: In mathematics:
xy
ux
n
uu
n
iii
n
iin
ii
10
1
1
1
ˆˆ
0ˆ
0
ˆ
thus,and 0ˆ
FIN357 Li 16
More terminology
SSR SSE SSTThen
(SSR) squares of sum residual theis ˆ
(SSE) squares of sum explained theis ˆ
(SST) squares of sum total theis
:following thedefine then Weˆˆ
part, dunexplainean and part, explainedan of up
made being asn observatioeach ofcan think We
2
2
2
i
i
i
iii
u
yy
yy
uyy
FIN357 Li 17
Notations Alert
The notation SSR (Sum of Squared Residuals) in this handout and my other lecture slides= ESS (Error Sum of Squares) in our textbook.
The notation SSE (Sum of Squared Explained) in this handout and my other lecture slides = RSS (Regressed Sum of Squares) in our textbook.
FIN357 Li 18
Goodness-of-Fit
How do we think about how well our sample regression line fits our sample data?
Can compute the fraction of the total sum of squares (SST) that is explained by the model, call this the R-squared of regression
R2 = SSE/SST = 1 – SSR/SST
FIN357 Li 19
OLS regressions
Now that we’ve derived the formula for calculating the OLS estimates of our parameters, you’ll be happy to know you don’t have to compute them by hand
Regressions in GRETL are very simple.
Have you installed the software yet?
FIN357 Li 20
Under some conditions, OLS esimated coefficients are unbiased.
22
1 1 2
21 1 1
. :
ˆ
ˆ 1 *
x i
i i
x
i ix
Define
S x x It could be shown that
x x u
S
E x x E uS
FIN357 Li 21
Unbiasedness Summary
The OLS estimates of 1 and 0 are unbiased
Remember unbiasedness is a description of the estimator – in a given sample we may be “near” or “far” from the true parameter
FIN357 Li 22
Variance of the OLS Estimators
Now we know that the sampling distribution of our estimated coefficient is centered around the true parameter
Want to think about how spread out this distribution is
Assume Var(u|x) =Var(u) = 2
FIN357 Li 23
Variance of OLS estimators
2 is called the error variance
, the square root of the error variance is called the standard deviation of the error
E(y|x)=0 + 1x and Var(y|x) = 2
FIN357 Li 24
Variance of OLS estimator
2
21̂x
VarS
FIN357 Li 25
Variance of OLS Summary
The larger the error variance, 2, the larger the variance of the slope estimate
The larger the variability in the xi, the smaller the variance of the slope estimate
Problem that the error variance is unknown
FIN357 Li 26
Estimating the Error Variance
We don’t know what the error variance, 2, is, because we don’t observe the errors, ui
What we observe are the residuals, ûi
We can use the residuals to form an estimate of the error variance
FIN357 Li 27
Estimating the Error Variance
0 1
2
2 2
ˆ ˆˆ
an unbiased estimator of is
1ˆ ˆ / 2
2
i i i
i
u y x
u SSR nn
FIN357 Li 28
Estimating Standard Error of coefficients Estimate
1
2 2
1̂ ˆ se / ix x
top related