lecture 4 this week’s reading: ch. 1 today: ch. 1: the simple regression model interpretation of...
TRANSCRIPT
Lecture 4
This week’s reading: Ch. 1
Today:Ch. 1: The Simple Regression Model• Interpretation of regression results• Goodness of fit
© Christopher Dougherty 1999–2006
XXnX1
Y
b1
XbbY
uXY
21
21
ˆ :line Fitted
:model True
1211̂ XbbY
1Y
b2
nY
nn XbbY 21ˆ
XbYb 21
We chose the parameters of the fitted line so as to minimize the sum of the squares of the residuals. As a result, we derived the expressions for b1 and b2.
22
XX
YYXXb
i
ii
DERIVING LINEAR REGRESSION COEFFICIENTS
© Christopher Dougherty 1999–2006
INTERPRETATION OF A REGRESSION EQUATION
The scatter diagram shows hourly earnings in 2002 plotted against years of schooling, defined as highest grade completed, for a sample of 540 respondents from the National Longitudinal Survey of Youth.
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© Christopher Dougherty 1999–2006
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Highest grade completed means just that for elementary and high school. Grades 13, 14, and 15 mean completion of one, two and three years of college.
INTERPRETATION OF A REGRESSION EQUATION
© Christopher Dougherty 1999–2006
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Grade 16 means completion of four-year college. Higher grades indicate years of postgraduate education.
INTERPRETATION OF A REGRESSION EQUATION
© Christopher Dougherty 1999–2006
. reg EARNINGS S
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 1, 538) = 112.15 Model | 19321.5589 1 19321.5589 Prob > F = 0.0000 Residual | 92688.6722 538 172.283777 R-squared = 0.1725-------------+------------------------------ Adj R-squared = 0.1710 Total | 112010.231 539 207.811189 Root MSE = 13.126
------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- S | 2.455321 .2318512 10.59 0.000 1.999876 2.910765 _cons | -13.93347 3.219851 -4.33 0.000 -20.25849 -7.608444------------------------------------------------------------------------------
This is the output from a regression of earnings on years of schooling, using Stata.
INTERPRETATION OF A REGRESSION EQUATION
© Christopher Dougherty 1999–2006
. reg EARNINGS S
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 1, 538) = 112.15 Model | 19321.5589 1 19321.5589 Prob > F = 0.0000 Residual | 92688.6722 538 172.283777 R-squared = 0.1725-------------+------------------------------ Adj R-squared = 0.1710 Total | 112010.231 539 207.811189 Root MSE = 13.126
------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- S | 2.455321 .2318512 10.59 0.000 1.999876 2.910765 _cons | -13.93347 3.219851 -4.33 0.000 -20.25849 -7.608444------------------------------------------------------------------------------
For the time being, we will be concerned only with the estimates of the parameters. The variables in the regression are listed in the first column and the second column gives the estimates of their coefficients.
INTERPRETATION OF A REGRESSION EQUATION
© Christopher Dougherty 1999–2006
. reg EARNINGS S
Source | SS df MS Number of obs = 540-------------+------------------------------ F( 1, 538) = 112.15 Model | 19321.5589 1 19321.5589 Prob > F = 0.0000 Residual | 92688.6722 538 172.283777 R-squared = 0.1725-------------+------------------------------ Adj R-squared = 0.1710 Total | 112010.231 539 207.811189 Root MSE = 13.126
------------------------------------------------------------------------------ EARNINGS | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- S | 2.455321 .2318512 10.59 0.000 1.999876 2.910765 _cons | -13.93347 3.219851 -4.33 0.000 -20.25849 -7.608444------------------------------------------------------------------------------
In this case there is only one variable, S, and its coefficient is 2.46. _cons, in Stata, refers to the constant. The estimate of the intercept is -13.93.
INTERPRETATION OF A REGRESSION EQUATION
© Christopher Dougherty 1999–2006
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Here is the scatter diagram again, with the regression line shown.
SEARNINGS 46.293.13 ^
INTERPRETATION OF A REGRESSION EQUATION
© Christopher Dougherty 1999–2006
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SEARNINGS 46.293.13 ^
What do the coefficients actually mean?
INTERPRETATION OF A REGRESSION EQUATION
© Christopher Dougherty 1999–2006
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SEARNINGS 46.293.13 ^
To answer this question, you must refer to the units in which the variables are measured.
INTERPRETATION OF A REGRESSION EQUATION
© Christopher Dougherty 1999–2006
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SEARNINGS 46.293.13 ^
S is measured in years (strictly speaking, grades completed), EARNINGS in dollars per hour. So the slope coefficient implies that hourly earnings increase by $2.46 for each extra year of schooling.
INTERPRETATION OF A REGRESSION EQUATION
© Christopher Dougherty 1999–2006
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SEARNINGS 46.293.13 ^
We will look at a geometrical representation of this interpretation. To do this, we will enlarge the marked section of the scatter diagram.
INTERPRETATION OF A REGRESSION EQUATION
© Christopher Dougherty 1999–2006
7
9
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15
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21
10.8 11 11.2 11.4 11.6 11.8 12 12.2
Years of schooling
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The regression line indicates that completing 12th grade instead of 11th grade would increase earnings by $2.46, from $13.07 to $15.53, as a general tendency.
One year
$2.46$13.07
$15.53
INTERPRETATION OF A REGRESSION EQUATION
© Christopher Dougherty 1999–2006
You should ask yourself whether this is a plausible figure. If it is implausible, this could be a sign that your model is misspecified in some way.
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SEARNINGS 46.293.13 ^
INTERPRETATION OF A REGRESSION EQUATION
© Christopher Dougherty 1999–2006
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SEARNINGS 46.293.13 ^
For low levels of education it might be plausible. But for high levels it would seem to be an underestimate.
INTERPRETATION OF A REGRESSION EQUATION
© Christopher Dougherty 1999–2006
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SEARNINGS 46.293.13 ^
What about the constant term? (Try to answer this question yourself before continuing with this sequence.)
INTERPRETATION OF A REGRESSION EQUATION
© Christopher Dougherty 1999–2006
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SEARNINGS 46.293.13 ^
Literally, the constant indicates that an individual with no years of education would have to pay $13.93 per hour to be allowed to work.
INTERPRETATION OF A REGRESSION EQUATION
© Christopher Dougherty 1999–2006
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SEARNINGS 46.293.13 ^
This does not make any sense at all. In former times craftsmen might require an initial payment when taking on an apprentice, and might pay the apprentice little or nothing for quite a while, but an interpretation of negative payment is impossible to sustain.
INTERPRETATION OF A REGRESSION EQUATION
© Christopher Dougherty 1999–2006
A safe solution to the problem is to limit the interpretation to the range of the sample data, and to refuse to extrapolate on the ground that we have no evidence outside the data range.
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INTERPRETATION OF A REGRESSION EQUATION
© Christopher Dougherty 1999–2006
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SEARNINGS 46.293.13 ^
With this explanation, the only function of the constant term is to enable you to draw the regression line at the correct height on the scatter diagram. It has no meaning of its own.
INTERPRETATION OF A REGRESSION EQUATION
© Christopher Dougherty 1999–2006
Another solution is to explore the possibility that the true relationship is nonlinear and that we are approximating it with a linear regression. We will soon extend the regression technique to fit nonlinear models.
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SEARNINGS 46.293.13 ^
INTERPRETATION OF A REGRESSION EQUATION
© Christopher Dougherty 1999–2006
Four useful results:
GOODNESS OF FIT
0e 0 iieXYY ˆ 0ˆ iieY
This sequence explains measures of goodness of fit in regression analysis. It is convenient to start by demonstrating three useful results. The first is that the mean value of the residuals must be zero.
© Christopher Dougherty 1999–2006
Four useful results:
0e 0 iieXYY ˆ 0ˆ iieY
The residual in any observation is given by the difference between the actual and fitted values of Y for that observation.
iiiii XbbYYYe 21ˆ
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
Four useful results:
0e 0 iieXYY ˆ 0ˆ iieY
First substitute for the fitted value.
iiiii XbbYYYe 21ˆ ii XbbY 21
ˆ
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
Four useful results:
0e 0 iieX
iiiii XbbYYYe 21ˆ
iii XbnbYe 21
YY ˆ 0ˆ iieY
Now sum over all the observations.
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
Four useful results:
0e 0 iieX
iiiii XbbYYYe 21ˆ
iii XbnbYe 21
0)( 22
21
XbXbYY
XbbYe
iii Xn
bbYn
en
11121
YY ˆ 0ˆ iieY
Dividing through by n, we obtain the sample mean of the residuals in terms of the sample means of X and Y and the regression coefficients.
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
Four useful results:
0e 0 iieX
iiiii XbbYYYe 21ˆ
iii XbnbYe 21
0)( 22
21
XbXbYY
XbbYe XbYb 21
If we substitute for b1, the expression collapses to zero.
iii Xn
bbYn
en
11121
YY ˆ 0ˆ iieY
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
Four useful results:
YY ˆ 0 iieX 0ˆ iieY0e
Next we will demonstrate that the mean of the fitted values of Y is equal to the mean of the actual values of Y.
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
Four useful results:
iii YYe ˆ
YY ˆ 0 iieX 0ˆ iieY0e
Again, we start with the definition of a residual.
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
Four useful results:
iii YYe ˆ
iii YYe ˆ
YY ˆ 0 iieX 0ˆ iieY
Sum over all the observations.
0e
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
Four useful results:
iii YYe ˆ
iii Yn
Yn
en
ˆ111
YYe ˆ
iii YYe ˆ
YY ˆ 0 iieX 0ˆ iieY
Divide through by n. The terms in the equation are the means of the residuals, actual values of Y, and fitted values of Y, respectively.
0e
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
Four useful results:
iii YYe ˆ
iii Yn
Yn
en
ˆ111
YYe ˆ YY ˆ
We have just shown that the mean of the residuals is zero. Hence the mean of the fitted values is equal to the mean of the actual values.
iii YYe ˆ
0e YY ˆ 0 iieX 0ˆ iieY
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
Four useful results:
0e YY ˆ 0 iieX 0ˆ iieY
Next we will demonstrate that the sum of the products of the values of X and the residuals is zero.
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
0221
21
iiii
iiiii
XbXbYX
XbbYXeX
Four useful results:
0e YY ˆ 0 iieX 0ˆ iieY
We start by replacing the residual with its expression in terms of Y and X.
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
Four useful results:
0e YY ˆ 0 iieX 0ˆ iieY
We expand the expression.
0221
21
iiii
iiiii
XbXbYX
XbbYXeX
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
Four useful results:
0e YY ˆ 0 iieX 0ˆ iieY
The expression is equal to zero. One way of demonstrating this would be to substitute for b1 and b2 and show that all the terms cancel out.
0221
21
iiii
iiiii
XbXbYX
XbbYXeX
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
Four useful results:
0e YY ˆ 0 iieX 0ˆ iieY
A neater way is to recall the first order condition for b2 when deriving the regression coefficients. You can see that it is exactly what we need.
02220 12
22
iiii XbYXXbbRSS
0221
21
iiii
iiiii
XbXbYX
XbbYXeX
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
Four useful results:
0e YY ˆ 0 iieX 0ˆ iieY
Finally we will demonstrate that the sum of the products of the fitted values of Y and the residuals is zero.
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
0
ˆ
21
21
21
ii
iii
iiii
eXbenb
eXbeb
eXbbeY
Four useful results:
0e YY ˆ 0 iieX 0ˆ iieY
We start by substituting for the fitted value of Y.
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
0
ˆ
21
21
21
ii
iii
iiii
eXbenb
eXbeb
eXbbeY
Four useful results:
0e YY ˆ 0 iieX 0ˆ iieY
We expand and rearrange.
enei
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
Four useful results:
0e YY ˆ 0 iieX 0ˆ iieY
The expression is equal to zero, given the first and third useful results.
0
ˆ
21
21
21
ii
iii
iiii
eXbenb
eXbeb
eXbbeY
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
222 ˆˆˆ
iiiii eYYeYeYYY
We now come to the discussion of goodness of fit. One measure of the variation in Y is the sum of its squared deviations around its sample mean, often described as the Total Sum of Squares, TSS.
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
iiiiii eYYYYe ˆˆ
222 ˆˆˆ
iiiii eYYeYeYYY
We will decompose TSS using the fact that the actual value of Y in any observations is equal to the sum of its fitted value and the residual.
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
iiiiii eYYYYe ˆˆ
222 ˆˆˆ
iiiii eYYeYeYYY
We substitute for Yi.
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
iiiiii eYYYYe ˆˆ
222 ˆˆˆ
iiiii eYYeYeYYY
YY ˆ 0e
From the useful results, the mean of the fitted values of Y is equal to the mean of the actual values. Also, the mean of the residuals is zero.
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
iiiiii eYYYYe ˆˆ
222 ˆˆˆ
iiiii eYYeYeYYY
Hence we can simplify the expression as shown.
YY ˆ 0e
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
iiiiii eYYYYe ˆˆ
222 ˆˆˆ
iiiii eYYeYeYYY
iiiii
iiiii
eYeYeYY
eYYeYYYY
2ˆ2ˆ
ˆ2ˆ
22
222
We expand the squared terms on the right side of the equation.
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
iiiiii eYYYYe ˆˆ
222 ˆˆˆ
iiiii eYYeYeYYY
iiiii
iiiii
eYeYeYY
eYYeYYYY
2ˆ2ˆ
ˆ2ˆ
22
222
We expand the third term on the right side of the equation.
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
iiiiii eYYYYe ˆˆ
222 ˆˆˆ
iiiii eYYeYeYYY
iiiii
iiiii
eYeYeYY
eYYeYYYY
2ˆ2ˆ
ˆ2ˆ
22
222
The last two terms are both zero, given the first and fourth useful results.
0ˆ iieY so ,0e
0 ie
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
iiiiii eYYYYe ˆˆ
222 ˆˆˆ
iiiii eYYeYeYYY
iiiii
iiiii
eYeYeYY
eYYeYYYY
2ˆ2ˆ
ˆ2ˆ
22
222
222 ˆiii eYYYY RSSESSTSS
Thus we have shown that TSS, the total sum of squares of Y can be decomposed into ESS, the ‘explained’ sum of squares, and RSS, the residual (‘unexplained’) sum of squares.
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
iiiiii eYYYYe ˆˆ
The words explained and unexplained were put in quotation marks because the explanation may in fact be false. Y might really depend on some other variable Z, and X might be acting as a proxy for Z. It would be safer to use the expression apparently explained instead of explained.
222 ˆˆˆ
iiiii eYYeYeYYY
iiiii
iiiii
eYeYeYY
eYYeYYYY
2ˆ2ˆ
ˆ2ˆ
22
222
222 ˆiii eYYYY RSSESSTSS
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
2
22
)(
)ˆ(
YY
YY
TSSESS
Ri
i
The main criterion of goodness of fit, formally described as the coefficient of determination, but usually referred to as R2, is defined to be the ratio of ESS to TSS, that is, the proportion of the variance of Y explained by the regression equation.
222 ˆiii eYYYY RSSESSTSS
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
Obviously we would like to locate the regression line so as to make the goodness of fit as high as possible, according to this criterion. Does this objective clash with our use of the least squares principle to determine b1 and b2?
2
22
)(
)ˆ(
YY
YY
TSSESS
Ri
i
222 ˆiii eYYYY RSSESSTSS
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
Fortunately, there is no clash. To see this, rewrite the expression for R2 in term of RSS as shown.
2
2
2
)(1
YY
e
TSSRSSTSS
Ri
i
2
22
)(
)ˆ(
YY
YY
TSSESS
Ri
i
222 ˆiii eYYYY RSSESSTSS
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
2
2
2
)(1
YY
e
TSSRSSTSS
Ri
i
2
22
)(
)ˆ(
YY
YY
TSSESS
Ri
i
The OLS regression coefficients are chosen in such a way as to minimize the sum of the squares of the residuals. Thus it automatically follows that they maximize R2.
222 ˆiii eYYYY RSSESSTSS
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
Another natural criterion of goodness of fit is the correlation between the actual and fitted values of Y. We will demonstrate that this is maximized by using the least squares principle to determine the regression coefficients
22
2
2
2
22
2
22ˆ,
ˆˆ
ˆ
ˆ
ˆ
ˆ
RYY
YY
YY
YY
YYYY
YY
YYYY
YYYYr
i
i
i
i
ii
i
ii
ii
YY
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
We will start with the numerator and substitute for the actual value of Y, and its mean, in the first factor.
22
2
2
2
22
2
22ˆ,
ˆˆ
ˆ
ˆ
ˆ
ˆ
RYY
YY
YY
YY
YYYY
YY
YYYY
YYYYr
i
i
i
i
ii
i
ii
ii
YY
2
2
ˆ
ˆˆ
ˆˆ
ˆˆˆ
YY
eYYeYY
YYeYY
YYeYeYYYYY
i
iiii
iii
iiiii
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
We rearrange a little.
22
2
2
2
22
2
22ˆ,
ˆˆ
ˆ
ˆ
ˆ
ˆ
RYY
YY
YY
YY
YYYY
YY
YYYY
YYYYr
i
i
i
i
ii
i
ii
ii
YY
2
2
ˆ
ˆˆ
ˆˆ
ˆˆˆ
YY
eYYeYY
YYeYY
YYeYeYYYYY
i
iiii
iii
iiiii
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
We expand the expression The last two terms are both zero (fourth and first useful results).
22
2
2
2
22
2
22ˆ,
ˆˆ
ˆ
ˆ
ˆ
ˆ
RYY
YY
YY
YY
YYYY
YY
YYYY
YYYYr
i
i
i
i
ii
i
ii
ii
YY
2
2
ˆ
ˆˆ
ˆˆ
ˆˆˆ
YY
eYYeYY
YYeYY
YYeYeYYYYY
i
iiii
iii
iiiii
so ,0e
0 ie
0ˆ iieY
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
Thus the numerator simplifies to the sum of the squared deviations of the fitted values.
22
2
2
2
22
2
22ˆ,
ˆˆ
ˆ
ˆ
ˆ
ˆ
RYY
YY
YY
YY
YYYY
YY
YYYY
YYYYr
i
i
i
i
ii
i
ii
ii
YY
2
2
ˆ
ˆˆ
ˆˆ
ˆˆˆ
YY
eYYeYY
YYeYY
YYeYeYYYYY
i
iiii
iii
iiiii
0ˆ iieY so ,0e
0 ie
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
We have the same expression in the denominator, under a square root. Cancelling, we are left with the square root in the numerator.
22
2
2
2
22
2
22ˆ,
ˆˆ
ˆ
ˆ
ˆ
ˆ
RYY
YY
YY
YY
YYYY
YY
YYYY
YYYYr
i
i
i
i
ii
i
ii
ii
YY
GOODNESS OF FIT
© Christopher Dougherty 1999–2006
22
2
2
2
22
2
22ˆ,
ˆˆ
ˆ
ˆ
ˆ
ˆ
RYY
YY
YY
YY
YYYY
YY
YYYY
YYYYr
i
i
i
i
ii
i
ii
ii
YY
Thus the correlation coefficient is the square root of R2. It follows that it is maximized by the use of the least squares principle to determine the regression coefficients.
GOODNESS OF FIT