Econ 444, Monday November 13, class 14

Transcription

1 Econ 444, Monday November 13, class 14 Robert de Jong 1 1 Department of Economics Ohio State University

2 Class program October 17 Return takehome # 4 Review of Monday s class New material: F -test

3 Fourth homework 1 Yes: > 1.96, so reject the null. 2 Yes: > 2.56, so reject the null. 3 t-value < A little bigger than 5%: < 1.73 < 1.96, so between 10% and 5% 5 No: 1.73 < 1.96, so we cannot reject. 6 No: 1.73 < 2.56, so we cannot reject.

4 Data mining In practice, we need to search for a decent specification Example: in wage equation, do we include geographic dummy variables, sector of employment, education of parents etc.? We should not be searching and searching for the best-looking equation Data mining: searching for the best specification by exhausting all possible equations that can be estimated Data mining in econometrics is a bad thing: this invalidates inference

5 Recommendation: Try a sensible specification first, and only change it if this is what the data suggest Example: geographic dummy variables are deemed irrelevant: leave them out, and perhaps try a regression with them included to see if they are significant Then if t-values suggest they are relevant, change the original plan

6 Olympic games and medals Question: which country did best? Data for: 1 Number of medals for every country in 2004 Olympics 2 GDP per capita for every country 3 Population How do we proceed?

7 1 What should be our Y i and what should be our X i? 2 Logarithms? 3 Once we have estimated the relationship, how do we identify the best performing county or countries? 4 What is the best performing country?

8 Best performing": country that, given gdp per capita and population, gets the most medals Best performing": farthest above the regression line Do we explain medals from population and gdp per capita? Probably high variance at high value for population Instead: explain medals per million (billion) inhabitants from gdp per capita Is this fair?

9 Perhaps explain medals per million (billion) inhabitants from gdp per capita and population, to compensate for the fact that larger countries can send limited number of athletes to Olympics Scope for the use of logarithms: use logarithm of nr. of medals or number of medals per billion inhabitants? This may not be appropriate here

10 F -test: tests a so-called multiple hypothesis" Examples: 1 β 1 = β 2 = β 3 = 0 (three regional dummies) 2 β 1 = β 2 (two income sources and expenditure on housing) 3 β 1 + β 2 = 1 4 β 1 = 0 5 All β i are zero, except for the intercept

11 Idea behind F -test When we do least squares, we minimize n (Y i β 0 β 1 X 1i... β K x Ki ) 2 i=1 and when we plug in the least squares estimator, we get RSS (residual sum of squares) Idea: minimize the sum of squares under the restriction of the F -test; resulting sum of squares: RRSS Now RSS RRSS; F -test compares these to sums of squares; sums of squares far apart suggests statistical evidence against the restriction Check p-value for F -test; test has F -distribution

12 Special cases: 1. Testing β 1 = 0 using F -test: F -test will be the square of the t-value, and the p-value for the F -test will be identical to the p-value for the t-test Conclusion: F -test for simple hypothesis unnecessary - same conclusion as t-test 2. Testing whether all β i are zero except for the constant: F -test for overall significance Reported in Eviews; nearly always rejects H 0 in practice F -test for overall significance in simple regression...?

13 Special cases: 1. Testing β 1 = 0 using F -test: F -test will be the square of the t-value, and the p-value for the F -test will be identical to the p-value for the t-test Conclusion: F -test for simple hypothesis unnecessary - same conclusion as t-test 2. Testing whether all β i are zero except for the constant: F -test for overall significance Reported in Eviews; nearly always rejects H 0 in practice F -test for overall significance in simple regression...?

14 Heteroskedasticity is the failure of model assumption 5 The model assumptions: 1 The regression model is linear in the coefficients, is correctly specified, and has an additive error term. 2 The error term has a zero population term. 3 All explanatory variables are uncorrelated with the error term. 4 Observations of the error term are uncorrelated with each other (no serial correlation). 5 The error term has a constant variance (no heteroskedasticity). 6 No explanatory variable is a perfect linear combination of any other explanatory variable(s) (no perfect multicollinearity). 7 The error term is normally distributed (this assumption is optional but usually is invoked).

15 Heteroskedasticity can take various forms Example: expenditure on housing vs. income Example: crop yield vs. world price over 24 years, technical innovation after 12 years Example: sales of VCRs in the period explained from price level and income Consequence of heteroskedasticity: 1 The coefficients will still remain correct. 2 t-values, standard errors and tests will be incorrect Compare to omitted variable or endogeneity: coefficients AND standard errors and tests will be incorrect

16 Testing: use the White test 1 Obtain residuals 2 Run a regression of the squared residuals on the regressors and on squares and cross-products. Example for two regressors: (e i ) 2 = α 0 + α 1 X 1i + α 2 X 2i +α 3 X 1i X 2i + α 3 X 2 1i + α 4X 2 2i + u i 3 Calculate nr 2 and compare to critical value from chi-square table with appropriate number of degrees of freedom l 4 l equals the number of variables in the above regression, not counting the constant (here: 5)

17 Problems with White test: 1 not automatic" - i.e. we get no p-value that can be compared to number of terms can be large if number of regressors is large

18 Remedies for heteroskedasticity 1 Get corrected t-values and standard errors: White s robust standard errors 2 Redefining the variables 3 Weighted least squares Solution 1: easy to implement (to be found in Eviews pull-down menu)

19 Redefining variables In the housing expenditure vs. income example: 1 use logarithms 2 regress housing expenditure as a fraction of income on the reciprocal of income

20 Implementing F -test in Eviews Dependent Variable: MATH10 Method: Least Squares Date: 11/22/04 Time: 09:48 Sample: Included observations: 408 Variable Coefficient Std. Error t-statistic Prob. C ENROLL LNCHPRG LSALARY R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Click: View - Wald-coefficient restrictions Wald Test: Equation: Untitled Null Hypothesis: C(3)=0 C(2)=0 F-statistic Probability Chi-square Probability