Prediction, Goodness-of-Fit, and Modeling Issues in Econometrics
Explore the concepts of prediction, goodness-of-fit, and modeling issues in econometrics through topics like least squares prediction, measuring goodness-of-fit, and variance analysis. Learn how factors such as uncertainty, sample size, and explanatory variable variation impact forecast error variance. Gain insights into point and interval predictions, with practical examples and illustrations.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.
E N D
Presentation Transcript
4.1 Least Squares Prediction 4.2 Measuring Goodness-of-Fit 4.3 Modeling Issues Principles of Econometrics, 3rd Edition Slide 4-2
= + + y 2 0 x e (4.1) 0 1 0 ( ) = + E y 2 0 x . We also assume that and ( ) 0 0 E e = ( ) 0 cov , 0 1,2, i e e i = = where e0 is a random error. We assume that and var e 0 1 ( ) = 2 0 , N = + y b 2 0 b x (4.2) 0 1 Principles of Econometrics, 3rd Edition Slide 4-3
Figure 4.1 A point prediction Principles of Econometrics, 3rd Edition Slide 4-4
( ) ( ) = = + + + y f y x e b b x (4.3) 0 0 1 2 0 0 1 2 0 ( ) ( ) E e ( ) E b ( ) E b x = + + + E f x 1 2 0 0 1 2 0 = + + + = 0 0 x x 1 2 0 1 2 0 2 ( ) x x x 1 N = + + 2 0 var( ) 1 f (4.4) 2 ( ) x i Principles of Econometrics, 3rd Edition Slide 4-5
The variance of the forecast error is smaller when the overall uncertainty in the model is smaller, as measured by the variance of the random errors ; the sample size N is larger; the variation in the explanatory variable is larger; and the value of is small. i. ii. iii. iv. Slide 4-6 Principles of Econometrics, 3rd Edition
2 ( ) x x x 1 N = + + 2 0 var( ) 1 f 2 ( ) x i ( ) f ( ) f = se var (4.5) ( ) f 0 y se t (4.6) c Principles of Econometrics, 3rd Edition Slide 4-7
y Figure 4.2 Point and interval prediction Principles of Econometrics, 3rd Edition Slide 4-8
= + = 83.4160 10.2096(20) + = y 287.6089 b b x 0 1 2 0 2 ( ) x x x 1 N = + + 2 0 var( ) 1 f 2 ( ) x i 2 2 N = + + 2 2 ( ) x x 0 2 ( ) x x i 2 N ( ) b = + + 2 2 ( ) var x x 0 2 = = 0 y se( ) 287.6069 2.0244(90.6328) 104.1323,471.0854 t f c Principles of Econometrics, 3rd Edition Slide 4-9
= + + y x e (4.7) 1 2 i i i = + ( ) E y y e (4.8) i i i = + y e y (4.9) i i i = + y e ( ) y y y (4.10) i i i 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
( ) 2 y N y 2 = i y 1 = + 2 2 2 i y e (4.11) ( ) ( ) y y y i i Principles of Econometrics, 3rd Edition Slide 4-12
= total sum of squares = SST: a measure of total variation in y about the sample mean. iy ( ) y 2 2 = 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. ( ) iy y 2 = 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. ie SST = SSR + SSE Principles of Econometrics, 3rd Edition Slide 4-13
SSR SST SSE SST = = 2 1 R (4.12) The closer R2 is to one, the closer the sample values yi are to the fitted regression equation . If R2= 1, then all the sample data fall exactly on the fitted 1 2 i i y b b x = + 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( , ) var( ) var( ) x x y xy = = (4.13) xy y x y xy = r (4.14) xy x y ( ) = ( )( ) 1 x x y y N xy i i ( ) = 2 ( ) 1 x x N (4.15) x i ( ) = 2 ( ) 1 y y N y i Principles of Econometrics, 3rd Edition Slide 4-15
= 2 2 xy r R = 2 2 R yy r 2measures the linear association, or goodness-of-fit, between the sample data R 2 is sometimes called a measure of and their predicted values. Consequently R goodness-of-fit. Principles of Econometrics, 3rd Edition Slide 4-16
( ) 2 = = 495132.160 SST y y i ( ) 2 = = = 2 i y e 304505.176 SSE y i i 304505.176 495132.160 SSE SST = = = 2 1 1 .385 R 478.75 xy = = = .62 r ( )( ) xy x 6.848 112.675 y Principles of Econometrics, 3rd Edition Slide 4-17
y Figure 4.4 Plot of predicted y, against y Principles of Econometrics, 3rd Edition Slide 4-18
FOOD_EXP = weekly food expenditure by a household of size 3, in dollars INCOME = weekly household income, in $100 units 83.42 10.21 + = 2 .385 FOOD_EXP= INCOME R * *** (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: = + + + + + + * 2 * = ( )( / ) x c = y x e c e x e 1 2 1 2 1 = = * 2 * where and c x x c 2 Changing the scale of y: = + + = + + * * 1 * 2 * / ( / ) c ( / ) c x ( / ) or e c y c y x e 1 2 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 logarithm is ln(x). The reciprocal: if x is a variable then its reciprocal is 1/x. Principles of Econometrics, 3rd Edition Slide 4-21
Figure 4.5 A nonlinear relationship between food expenditure and income Principles of Econometrics, 3rd Edition Slide 4-22
The log-log model = + ln( ) ln( ) y x 1 2 The parameter is the elasticity of y with respect to x. The log-linear model = + ln( ) y x 1 2 i i A one-unit increase in x leads to (approximately) a 100 2 percent change in y. The linear-log model y ( ) x = + = 2 2ln or 100 y ( ) 1 100 x x A 1% increase in x leads to a 2/100 unit change in y. Principles of Econometrics, 3rd Edition Slide 4-23
The reciprocal model is 1 = + + _ FOOD EXP e 1 2 INCOME The linear-log model is = + + _ ln( ) FOOD EXP INCOME e 1 2 Principles of Econometrics, 3rd Edition Slide 4-24
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 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. 2. 3. Principles of Econometrics, 3rd Edition Slide 4-25
