12/10/2007
Prediction, Goodness-of-Fit, and Modeling Issues
Prepared by Vera Tabakova, East Carolina University
4.1 Least Squares Prediction
4.2 Measuring Goodness-of-Fit
4.3 Modlideling Issues
4.4 Log-Linear Models
Principles of Econometrics, 3rd Edition Slide 4-2
yxe01200=+ββ + (4.1)
where e0 is a random error. We assume that Ey ( 0120 ) =β +β x and . We also ass ume that 2 and E (e0 ) = 0 var (e0 ) =σ
cov(ee0 ,i ) == 0 i 1,2,… , N
ybbxˆ0120=+ (4.2)
Principles of Econometrics, 3rd Edition Slide 4-3
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Figure 4.1 A point prediction
Principles of Econometrics, 3rd Edition Slide 4-4
f =−=β+β+yy00ˆ ( 1200 xe) −( bbx 120 + ) (4.3)
Ef( ) = β+β120 x + Ee( 0) − ⎣⎦⎡+ ⎡ Eb( 1) + Ebx( 20) ⎤
=β120 +βxx +00 −[] β 120 +β =
2 2 ⎡⎤1 ()xx0 − var(f )=σ⎢⎥ 1 + + 2 (4.4) ⎣⎦Nxx∑()i −
Principles of Econometrics, 3rd Edition Slide 4-5
The variance of the forecast error is smaller when i. the overall uncertainty in the model is smaller, as measured by the variance of the random errors ; ii. the sample size N is larger; iii. the variation in the exppylanatory variable is larg g;er; and iv. the value of is small.
Principles of Econometrics, 3rd Edition Slide 4-6
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⎡⎤2 2 1(xx0 − ) var(f )=σˆ ⎢⎥ 1 + + 2 ⎣⎦Nxx∑()i −
se()f = var ()f (4.5)
ytˆ0 ± cse( f) (4.6)
Principles of Econometrics, 3rd Edition Slide 4-7
y
Figure 4.2 Point and interval prediction
Principles of Econometrics, 3rd Edition Slide 4-8
ybbxˆ0120=+ =83.4160 + 10.2096(20) = 287.6089
⎡⎤2 2 1(xx0 − ) var(f )=σˆ ⎢⎥ 1 + + 2 ⎣⎦Nxx∑()i −
22 22σσˆˆ =σˆ + +()xx0 − 2 N ∑()xxi −
σˆ 2 =σˆ 22 + +()varxx − () b N 02
ytˆ0 ±=cse( f ) 287.6069 ± 2.0244(90.6328) =[ 104.1323,471.0854]
Principles of Econometrics, 3rd Edition Slide 4-9
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yxeiii=β12 +β + (4.7)
yEyeiii=+() (4.8)
yyeiii=+ˆˆ (4.9)
yyiii−=() yyeˆˆ − + (4.10)
Principles of Econometrics, 3rd Edition Slide 4-10
Figure 4.3 Explained and unexplained components of yi
Principles of Econometrics, 3rd Edition Slide 4-11
()yy− 2 σ=ˆ 2 ∑ i y N −1
222(4.11) ∑∑∑()()yyiii−= yyˆˆ −+ e
Principles of Econometrics, 3rd Edition Slide 4-12
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2 ∑()yi −y = total sum of squares = SST: a measure of total variation in y about the sample mean.
2 ∑()yˆi − y = sum of squares due to the regression = SSR: that part of total variation in y, about the sample mean, that is explained by, or due to, the regression. Also known as the “explained sum of squares .”
2 ∑eˆi = sum of squares due to error = SSE: that part of total variation in y about its mean that is not explained by the regression. Also known as the unexplained sum of squares, the residual sum of squares, or the sum of squared errors.
SST = SSR + SSE
Principles of Econometrics, 3rd Edition Slide 4-13
2 SSR SSE R ==−1 (4.12) SST SST
2 The closer R is to one, the closer the sample values yi are to the fitted regression 2 equation ybbxˆ ii =+ 12 . If R = 1, then all the sample data fall exactly on the fitted least squares line, so SSE = 0, and the model fits the data “perfectly.” If the sample data for y and x are uncorrelated and show no linear association, then the least squares fitted line is “horizontal,” so that SSR = 0 and R2 = 0.
Principles of Econometrics, 3rd Edition Slide 4-14
cov(xy , ) σxy ρ=xy = (4.13) var(xy ) var( ) σσx y
σˆ xy rxy = (4.14) σσˆˆx y
σ=ˆ xy∑()()xxyy i − i −( N − 1)
2 σ=ˆ xi∑()xx −() N − 1 (4.15)
2 σ=ˆ yi∑()yy −() N − 1
Principles of Econometrics, 3rd Edition Slide 4-15
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22 rRxy =
22 R = ryyˆ
2 R measures the linear association, or goodness-of-fit, between the sample data 2 and their predicted values. Consequently R is sometimes called a measure of “goodness-of-fit.”
Principles of Econometrics, 3rd Edition Slide 4-16
2 SST=−=∑() yi y 495132.160
2 2 SSE=−==∑∑() yii yˆˆ e i304505.176
SSE 304505.176 R2 =−11 =− = .385 SST 495132.160
σˆ xy 478.75 rxy == =.62 σσˆˆxy ()()6.848 112.675
Principles of Econometrics, 3rd Edition Slide 4-17
Figure 4.4 Plot of predicted y, yˆ against y
Principles of Econometrics, 3rd Edition Slide 4-18
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FOOD_EXP = weekly food expenditure by a household of size 3, in dollars
INCOME = weekly household income, in $100 units
FOOD_EXP =83.42+= 10.21 INCOME R2 .385 (se) (43.41)**** (2.09)
* indicates significant at the 10% level ** indicates significant at the 5% level *** indicates significant at the 1% level
Principles of Econometrics, 3rd Edition Slide 4-19
4.3.1 The Effects of Scaling the Data
Changing the scale of x:
** yxecxcexe= β12+ β + =()(/)β 1+ β 2+ =β 12+ β +
** where ββ22==cxxc and
Changing the scale of y:
*** * yc/(/)(/)(/)=β12 c +β cx + ec or y =β+β 12 x + e
Principles of Econometrics, 3rd Edition Slide 4-20
Variable transformations:
Power: if x is a variable then xp means raising the variable to the power p; examples are quadratic (x2) and cubic (x3) transformations.
The natural logarithm: if x is a variable then its natural logg(arithm is ln(x).
The reciprocal: if x is a variable then its reciprocal is 1/x.
Principles of Econometrics, 3rd Edition Slide 4-21
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Figure 4.5 A nonlinear relationship between food expenditure and income
Principles of Econometrics, 3rd Edition Slide 4-22
The log-log model
ln(yx )=β12 +β ln( ) The parameter β is the elasticity of y with respect to x.
The log-linear model
ln(yxii ) =β12 +β
A one-unit increase in x leads to (approximately) a 100 β2 percent change in y.
The linear-log model Δβy yx=β +βln() or = 2 12 100()Δ xx 100
A 1% increase in x leads to a β2/100 unit change in y.
Principles of Econometrics, 3rd Edition Slide 4-23
The reciprocal model is 1 FOOD_ EXP= β + β + e 12INCOME
The linear-log model is
FOOD_ln() EXP=β12 +β INCOME + e
Principles of Econometrics, 3rd Edition Slide 4-24
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Remark: Given this array of models, that involve different transformations of the dependent and independent variables, and some of which have similar shapes, what are some ggguidelines for choosing a functional form? 1. Choose a shape that is consistent with what economic theory tells us about the relationship. 2. Choose a shape that is sufficiently flexible to “fit” the data 3. Choose a shape so that assumptions SR1-SR6 are satisfied, ensuring that the least squares estimators have the desirable properties described in Chapters 2 and 3.
Principles of Econometrics, 3rd Edition Slide 4-25
Figure 4.6 EViews output: residuals histogram and summary statistics for food expenditure example
Principles of Econometrics, 3rd Edition Slide 4-26
The Jarque-Bera statistic is given by
2 N ⎛⎞()K − 3 JB=+⎜⎟ S 2 ⎜⎟ 64⎝⎠ where N is the sample size, S is skewness, and K is kurtosis.
In the food expenditure example
2 40 ⎛⎞()2.99− 3 JB =−⎜⎟.0972 + = .063 ⎜⎟ 64⎝⎠
Principles of Econometrics, 3rd Edition Slide 4-27
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Figure 4.7 Scatter plot of wheat yield over time
Principles of Econometrics, 3rd Edition Slide 4-28
YIELDttt=β12 +β TIME + e
2 YIELDtt=+.638 .0210 TIME R = .649 (se) (.064) (.0022)
Principles of Econometrics, 3rd Edition Slide 4-29
Figure 4.8 Predicted, actual and residual values from straight line
Principles of Econometrics, 3rd Edition Slide 4-30
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Figure 4.9 Bar chart of residuals from straight line
Principles of Econometrics, 3rd Edition Slide 4-31
3 YIELDttt=β12 +β TIME + e
TIMECUBE= TIME3 1000000
2 YIELDtt=+0.874 9.68 TIMECUBE R = 0.751 (se) (.036) (.082)
Principles of Econometrics, 3rd Edition Slide 4-32
Figure 4.10 Fitted, actual and residual values from equation with cubic term
Principles of Econometrics, 3rd Edition Slide 4-33
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4.4.1 The Growth Model
ln(YIELDt ) =++ ln( YIELD0 ) ln( 1 g) t
= β+β12t
ln()YIELDt =− .3434 + .0178 t (se) (.0584) (.0021)
Principles of Econometrics, 3rd Edition Slide 4-34
4.4.2 A Wage Equation
ln(WAGE) =++ ln( WAGE0 ) ln( 1 r) EDUC
= βββ12+ β EDUC
ln ()WAGE=+× .7884 .1038 EDUC (se) (.0849) (.0063)
Principles of Econometrics, 3rd Edition Slide 4-35
4.4.3 Prediction in the Log-Linear Model
yybbxˆ ==+exp ln exp n ( ()) ()12
22σˆ 2 yEˆˆˆcn==Ebb( y) exp(bb12 ++σ=xye2)
ln ()WAGE=+× .7884 .1038 EDUC =+×= .7884 .1038 12 2.0335
σˆ 2 2 yEyyeˆˆcn==() =7.6408 × 1.1276 = 8.6161
Principles of Econometrics, 3rd Edition Slide 4-36
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4.4.4 A Generalized R2 Measure
2 22ˆ Ryyrgyy=⎡⎣⎦corr() , ⎤ = , ˆ
2 22ˆ Ryygc=⎡⎣⎦corr() , ⎤ = .4739 = .2246
R2 values tend to be small with microeconomic, cross-sectional data, because the variations in individual behavior are difficult to fully explain.
Principles of Econometrics, 3rd Edition Slide 4-37
4.4.5 Prediction Intervals in the Log-Linear Model
⎡⎤expln ()y −+tf se() ,expln () ytf se() ⎣⎦⎢⎥( cc) ( )
⎣⎦⎡−×exp( 2.0335 1.96 .4905) ,exp( 2.0335 +×⎤= 1.96 .4905) [ 2.9184,20.0046]
Principles of Econometrics, 3rd Edition Slide 4-38
coefficient of linear model determination linear relationship correlation linear-log model data scale log-linear model forecast error log-log model forecast standard log-normal error distribution functional form prediction goodness-of-fit prediction interval growth model R2 Jarque-Bera test residual kurtosis skewness least squares predictor
Principles of Econometrics, 3rd Edition Slide 4-39
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Principles of Econometrics, 3rd Edition Slide 4-40
f =−=β+β+yy00ˆ ( 1200 xe) −+( bbx 120)
2 var( yˆ0120102012) =+=+ var(bbx) var( b) x var( b) + 2 x cov( bb , )
22 2 σ ∑ xi 22σ−x =+22xx00 +σ2 2 Nxx∑∑()ii−−() xx ∑() xx i −
Principles of Econometrics, 3rd Edition Slide 4-41
⎡⎤⎡⎤22 ⎧⎫22 222 ⎧⎫ 22 σ ∑ xi ⎪⎪σσNx x0 σ−()2xx0 ⎪⎪ σ Nx var ()yˆ0 =−⎢⎥⎢⎥⎨⎬ +++ ⎨⎬ N xx−−−−−22222 N xx xx xx N xx ⎣⎦⎣⎦⎢⎥⎢⎥∑∑∑∑∑()iiiii⎩⎭⎪⎪() () () ⎩⎭⎪⎪()
⎡⎤2222 2 ∑ xNxi − xxxx00−+2 =σ⎢⎥22 + ⎣⎦⎢⎥Nxx∑∑( ii−−) ( xx)
⎡⎤22 2 ∑()xxi −−() xx0 =σ⎢⎥22 + ⎣⎦⎢⎥Nxx∑∑()ii−−() xx
2 ⎡⎤1 ()xx− =σ2 ⎢⎥ + 0 N 2 ⎣⎦⎢⎥∑()xi − x
Principles of Econometrics, 3rd Edition Slide 4-42
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f ~(0,1)N var(f )
2 2 ⎡⎤1(xx0 − ) var( f ) =σˆ ⎢⎥ 1 + + 2 ⎣⎦Nxx∑()i −
fyy00− ˆ = ~ t(2)N− (4A.1) var ()f se(f )
(4A.2) Pt()1−≤≤cc tt =−α
Principles of Econometrics, 3rd Edition Slide 4-43
yy− ˆ Pt[]1−≤00 ≤ t =−α ccse(f )
Py[ ˆˆ000−≤≤+=−α tccse( f ) y y t se( f )] 1
Principles of Econometrics, 3rd Edition Slide 4-44
2 2 22 ()yiiiiiii−=−+=−++−yyyeyyeyye[()ˆˆˆˆˆˆ] () 2()
2 22 ∑∑∑∑( yiiiii−=yyyeyye) ()ˆˆˆˆ −++ 2()−
∑∑∑∑∑( yˆˆˆˆˆii−=ye) yeye iii − =( bbxeye12 + iii) ˆ − ˆ
=+bebxeye12∑∑ˆˆˆiiii − ∑
Principles of Econometrics, 3rd Edition Slide 4-45
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∑∑eybbxyNbbxˆii=−−=−−=( 12 ii) ∑ 12 ∑ i0
2 ∑∑xeiiˆ =−−=−−= x i( y i b12 b x i) ∑ xy i i b 1 ∑∑ x i b 2 x i 0
∑( yyeˆˆii−=) 0
If the model contains an intercept it is guaranteed that SST = SSR + SSE.
If, however, the model does not contain an intercept, then ∑ e ˆ i ≠ 0 and SST ≠ SSR + SSE.
Principles of Econometrics, 3rd Edition Slide 4-46
Suppose that the variable y has a normal distribution, with mean μ and variance σ2. If we consider we = y then ywN =μσ ln ( ) ~ ( , 2 ) is said to have a log-normal distribution.
2 E (we) = μ+σ 2
22 var()we=−2μ+σ( e σ 1)
Principles of Econometrics, 3rd Edition Slide 4-47
Given the log-linear model ln ()y =β12 +βxe +
If we assume that eN~0,( σ2 )
βββ12+β xii+exeβ ββ 12+β i i Ey( i ) === Ee( ) Ee( e)
2 2 eEeeeβ+β12xeii()== β+β 12 x iσ 2 e β+β 12 x i +σ 2
2 bbx12++σi ˆ 2 Ey()i = e
Principles of Econometrics, 3rd Edition Slide 4-48
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The growth and wage equations:
β2 β=2 ln( 1 +r) and re=−1
bN~var~ ,var b2 xx2 222(β=σ( ) ∑( i − ) )
b β+var(b ) /2 bb22+var() /2 2 22 ˆ Ee⎣⎦⎡⎤= e re=−1
2 2 var()bxx2 =σˆ ∑()i −
Principles of Econometrics, 3rd Edition Slide 4-49
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