4.1 Least Squares Prediction 4.2 Measuring Goodness-of-Fit. 4.3 Modeling Issues. 4.4 Log-Linear Models

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2 4.1 Least Squares Prediction 4. Measuring Goodness-of-Fit 4.3 Modeling Issues 4.4 Log-Linear Models

3 y = β + β x + e ( ) E y where e 0 is a random error. We assume that and E( e 0 ) = 0 var ( e 0 ) = σ cov ( e, e ) = 0 i= 1,, K, N 0 i. We also assume that and ŷ0 = b1+ bx0 = β+β x 0 1 0

4 Figure 4.1 A point prediction Principles of Econometrics, 3rd Edition Slide 4-4

5 ( ) ( ) f = y yˆ = β +β x + e b + b x ( ) ( ) ( ) ( ) E f =β 1 +β x0 + E e0 E b1 + E b x0 [ x ] =β +β x + 0 β +β = ( x0 x) var( f ) =σ N ( xi x)

6 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 explanatory variable is larger; and iv. the value of is small.

7 1 ( x0 x) var( f ) =σ ˆ N ( xi x) se ( f ) = var ( f ) ( ) yˆ 0 ± tcse f

8 y Figure 4. Point and interval prediction Principles of Econometrics, 3rd Edition Slide 4-8

9 yˆ = b + b x = (0) = ( x0 x) var( f ) =σ ˆ N ( xi x) 0 σ垐 =σ ˆ + + ( ) x0 x N ( xi x) σˆ =σ ˆ + + ( ) var N σ x0 x b ( ) [ ] yˆ ± t se( f) = ±.044(90.638) = , c

10 y = β +β x + e i 1 i i y = E( y ) + e i i i y = y垐 + e i i i y y = ( y垐 y) + e i i i

11 Figure 4.3 Explained and unexplained components of y i Principles of Econometrics, 3rd Edition Slide 4-11

12 σ ˆ = y ( y ) i y N 1 ( y y) = ( y垐 y) + e i i i

13 ( yi y) = total sum of squares = SST: a measure of total variation in y about the sample mean. ( yˆ i 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. 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

14 R SSR = = 1 SST SSE SST The closer R is to one, the closer the sample values y i are to the fitted regression y = b + bx equation. If R = 1, then all the sample data fall exactly on the fitted ˆi 1 i 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 R = 0.

15 cov( xy, ) σxy ρ xy = = var( x) var( y) σσ x y r xy σˆ xy = σ 垐 σ x y ( ) ˆ x ( xi x) N 1 ( ) ˆ y ( yi y) N 1 ( ) σ ˆ = ( x x)( y y) N 1 xy i i σ = σ =

16 r = R xy R = r yy ˆ R measures the linear association, or goodness-of-fit, between the sample data and their predicted values. Consequently R is sometimes called a measure of goodness-of-fit.

17 ( i ) SST = y y = ( ) i i i SSE = y y垐 = e = SSE R = 1 = 1 =.385 SST r xy σˆ xy = = =.6 σσ 垐 x y ( )( )

18 Figure 4.4 Plot of predicted y, ŷ against y Principles of Econometrics, 3rd Edition Slide 4-18

19 FOOD_EXP = weekly food expenditure by a household of size 3, in dollars INCOME = weekly household income, in $100 units FOOD_EXP = INCOME R =.385 (se) (43.41) (.09) * *** * indicates significant at the 10% level ** indicates significant at the 5% level *** indicates significant at the 1% level

20 4.3.1 The Effects of Scaling the Data Changing the scale of x: y =β +β x+ e = β + ( cβ )( x/ c) + e = β +β x + e where β * * = cβ and x = x c * * Changing the scale of y: y/ c= ( β / c) + ( β / c) x+ ( e/ c) or y =β +β x+ e * * * * 1 1

21 Variable transformations: Power: if x is a variable then x p means raising the variable to the power p; examples are quadratic (x ) and cubic (x 3 ) transformations. The natural logarithm: if x is a variable then its natural logarithm is ln(x). The reciprocal: if x is a variable then its reciprocal is 1/x.

22 Figure 4.5 A nonlinear relationship between food expenditure and income Principles of Econometrics, 3rd Edition Slide 4-

23 The log-log model ln( y) = β +β ln( x) 1 The parameter β is the elasticity of y with respect to x. The log-linear model ln( y ) = β +β i 1 x i A one-unit increase in x leads to (approximately) a 100 β percent change in y. The linear-log model ln ( ) or Δy β y =β 1 +β x = 100( Δ xx) 100 A 1% increase in x leads to a β /100 unit change in y.

24 The reciprocal model is 1 FOOD _ EXP =β 1+β + e INCOME The linear-log model is FOOD _ EXP =β +β ln( INCOME) + e 1

25 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 guidelines for choosing a functional form? 1.Choose a shape that is consistent with what economic theory tells us about the relationship..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 and 3.

26 Figure 4.6 EViews output: residuals histogram and summary statistics for food expenditure example Principles of Econometrics, 3rd Edition Slide 4-6

27 The Jarque-Bera statistic is given by JB ( K ) N = S where N is the sample size, S is skewness, and K is kurtosis. In the food expenditure example JB ( ) = =

28 Figure 4.7 Scatter plot of wheat yield over time Principles of Econometrics, 3rd Edition Slide 4-8

29 YIELD =β +β TIME + e t 1 t t YIELD = + TIME R = t t.649 (se) (.064) (.00)

30 Figure 4.8 Predicted, actual and residual values from straight line Principles of Econometrics, 3rd Edition Slide 4-30

31 Figure 4.9 Bar chart of residuals from straight line Principles of Econometrics, 3rd Edition Slide 4-31

32 YIELD =β +β TIME + e 3 t 1 t t TIMECUBE = TIME YIELD = + TIMECUBE R = t t (se) (.036) (.08)

33 Figure 4.10 Fitted, actual and residual values from equation with cubic term Principles of Econometrics, 3rd Edition Slide 4-33

34 4.4.1 The Growth Model ( ) = ( ) + ( + ) ln YIELD ln YIELD ln 1 g t t =β +β 1 t 0 ( YIELD ) ln = t t (se) (.0584) (.001)

35 4.4. A Wage Equation ( ) = ( ) + ( + ) ln WAGE ln WAGE ln 1 r EDUC =β +β 1 0 EDUC ( WAGE) ln = EDUC (se) (.0849) (.0063)

36 4.4.3 Prediction in the Log-Linear Model ( ( )) ( 1 ) yˆ = exp ln y = exp b + b x n ( ) exp( 1 ) y垐 = E y = b + b x+σ = y? e σ c ( WAGE) ˆ n ln = EDUC = =.0335 ( ) ˆ y垐 = E y = y e σ = = c n

37 4.4.4 A Generalized R Measure ( ˆ ) R = corr yy, = r R g y, yˆ ( y yˆ ) = corr, =.4739 =.46 g c R values tend to be small with microeconomic, cross-sectional data, because the variations in individual behavior are difficult to fully explain.

38 4.4.5 Prediction Intervals in the Log-Linear Model ( ( y) t ( )) c f ( y) + tc ( f ) ( ) exp ln se,exp ln se ( ) ( ) [ ] exp ,exp =.9184,0.0046

39 Principles of Econometrics, 3rd Edition Slide 4-39

40 Principles of Econometrics, 3rd Edition Slide 4-40

41 ( ) ( ) f = y yˆ = β +β x + e b + b x ( yˆ ) = ( b + b x ) = ( b ) + x ( b ) + x ( b b ) var var var var cov, σ xi σ x x0 x0 ( i ) ( i ) ( i ) = + + σ N x x x x x x Principles of Econometrics, 3rd Edition Slide 4-41

42 var ( yˆ ) σ x ( i σ Nx σ x σ xx 0 0 ) σ Nx ( i ) ( i ) ( i ) ( i ) ( i ) = N x x N x x x x x x N x x 0 xi Nx x x x + x =σ + N ( xi x) ( xi x) 0 0 ( i ) ( i ) ( x x) ( x ) i x ( ) ( i ) x x x x =σ + N x x x x 0 1 =σ + N 0 Principles of Econometrics, 3rd Edition Slide 4-4

43 f var( f ) ~ N(0,1) 1 ( x0 x) var ( f ) =σ ˆ N ( xi x) f y yˆ = var se( f ) ( f ) 0 0 ~ t ( N ) (4A.1) P( t t t ) = 1 α c c (4A.) Principles of Econometrics, 3rd Edition Slide 4-43

44 y ˆ 0 y0 P[ tc tc] = 1 α se( f ) [ ] P y垐 t se( f) y y + t se( f) = 1 α 0 c 0 0 c Principles of Econometrics, 3rd Edition Slide 4-44

45 ( ) [ ] ( 垐 ) ( 垐 ) ( 垐 i = i + i = i + i + i ) i y y y y e y y e y y e ( ) 垐 y y = ( y y) + e + ( y垐 y) e i i i i i ( ) = = ( 1+ ) y垐 y e ye 垐 y e垐 b b x e y e? i i i i i i i i = b e垐 + b xe y e? 1 i i i i Principles of Econometrics, 3rd Edition Slide 4-45

46 ( 1 ) 1 eˆ = y b b x = y Nb b x = 0 i i i i i xe i i xi( yi b1 bxi) xy i i b1 xi b xi ˆ = = = 0 ( yi ) 垐 y e = 0 i If the model contains an intercept it is guaranteed that SST = SSR + SSE. If, however, the model does not contain an intercept, then eˆ and i 0 SST SSR + SSE. Principles of Econometrics, 3rd Edition Slide 4-46

47 Suppose that the variable y has a normal distribution, with mean μ and variance σ. y If we consider w= e then y= ln ( w) ~ N( μσ, ) is said to have a log-normal distribution. E( w) e μ+σ = ( w) e μ+σ ( σ = e ) var 1 Principles of Econometrics, 3rd Edition Slide 4-47

48 Given the log-linear model If we assume that ( ) e~ N 0, σ ln ( y) = β+β 1 x+ e ( ) ( β+β ) ( ) 1 xi+ ei β+β 1 xi ei i E y = E e = E e e = ( ) 1 xi ei 1 xi σ 1 xi e E e = e e = e β+β β+β β+β +σ ( ) E y = e + +σ i b1 bx i ˆ Principles of Econometrics, 3rd Edition Slide 4-48

49 The growth and wage equations: ln ( 1 r) β = + and r = e β 1 ( ( ) ( ) ) β =σ b ~ N,var b xi x = b E e e β + ( b ) var / ( b ) b var / ˆ 1 r = e + var ( b ) ˆ ( ) =σ xi x Principles of Econometrics, 3rd Edition Slide 4-49