Applied Econometrics. Professor Bernard Fingleton

Transcription

1 Applied Econometrics Professor Bernard Fingleton

2 Regression A quick summary of some key issues

3 Some key issues Text book JH Stock & MW Watson Introduction to Econometrics 2nd Edition Software Gretl Gretl.sourceforge.net 3

4 Course outline Week 5 introduction to regression Week 6 endogeneity & instrumental variables Week 7 panel data Week 8 spurious regression, Dickey-Fuller etc Week 9 cointegration and error correction Week 10 - autoregressive distributed lag models Week 11 vector autoregression (VAR), vector error correction, multiple cointegrating vectors Week 12 VAR, Johansen etc 4

5 Regression Regression is used to analyze how a single dependent variable (or Y variable) is affected by the values of one or more independent variables (also called regressors, X variables, factors). 5

6 6 Multiple regression ˆ... ˆ ˆ ˆ ˆ... ) ( = = = = k k k k k k X b X b X b b Y X b X b X b b Y E e X b X b X b b Y e X b X b b Y b 1 is change in E(Y) per unit change in X 1 b k-1 is change in E(Y) per unit change in X k-1

7 Interpreting partial regression coefficients b i is the change in E(Y) per unit change in X i, with all other variables held statistically constant E(Y) = b 0 + b 1 X 1 + b 2 X 2 assume we change X 1 by an amount equal to ΔX 1, but keep X 2 constant; this changes E(Y) to new E(Y) new E(Y) = b 0 + b 1 (X 1 + ΔX 1 ) + b 2 X 2 new E(Y) - E(Y) = ΔΕ( Y) = b 1 ΔX 1 thus if ΔX 1 = 1, ΔΕ( Y) = b 1 7

8 Theory indicating X variables output = f(labour, capital) adopt a Cobb-Douglas production function output = labour α capital β ln(output) = α ln(labour) + β ln(capital) if α + β > 1 we have increasing returns doubling inputs more than doubles output if α + β = 1 we have constant returns to scale 9

9 dy / Y elasticity = ;% change in Y per 1% change in X dx / X log Yˆ = b ˆ + bˆ log X 0 1 Yˆ = exp( bˆ ) X = b ˆ X bˆ 0 0 bˆ 1 1 dyˆ ˆ ˆ ˆ b1 1 = b 0 b1x dx dyˆ ˆ ˆ b1 = b1( b 0 X ) X dx dyˆ ˆ ˆ 1 = byx 1 dx ˆ / ˆ ˆ dy Y b1 = dx / X ˆ 1

10 Data indicating X variables letting the data speak is a good way to obtain a realistic theory we look at the data to identify important variables unimportant variables, that are indistinguishable from random variation and can be left in the error term 18

11 Methods for choosing X variables R 2 t tests F tests 19

12 R R = 1 - eˆ / S ˆ YY = corr(y,y ) 2 indicates, on a scale from 0 to 1, or 0% to 100% how much of Y s s variation is accounted for by the Xs s contained in the regression model [this equation assumes that b 0 is present] 20

13 R 2 DISADVANTAGES R 2 s s probability distribution is not constant making it difficult to objectively compare the R 2 of different models. R 2 ALWAYS increases if additional (perhaps unimportant)variables are added to the model. Hence the most complex model always seems the best using R 2. BUT R 2 -adjusted takes into account the number of Xs. 21

14 t -test Say we wish to test whether a particular variable, X i, should be included [In practice i could be,say, 2 if we were testing X 2 ] H 0 : b i =0 [X i has no effect on Y] t bˆ b bˆ = i i = i σˆ s se..( bˆ ) XX t ratio ~ t T k i When H o correct for population T = sample size, k = number of regression coefficients 22

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16 Taiwanese agricultural output The regression equation is ln output = ln labour ln capital Predictor Coef Stdev t-ratio p Constant b 0 = ln labour b 1 = ln capital b 2 = For ln(labour) from t tables t = / = 2.78 t crit = 2.18 with (T-k)( ) = = 12 degrees of freedom, crit is the value (ignoring sign) with p-value p = 0.05 in the 12 distribution t crit t 12 since t > t crit (ie p-value for t = < 0.05) reject H o that b 1 =0

17 Analysis of Variance (ANOVA) 25

18 F test : Jointly testing a group of X s H : b = b =...= b = O k 1 The test statistic is the F ratio calculated from an ANOVA table This is calculated automatically whenever a regression model is fitted to a data set 26

19 F test of a group of X variables H : b = b =... b = 0 H O A 1 2 k 1 is that H is untrue O ( SYY D)/( k 1) F = ~ Fk 1, T k assuming HO is true D/( T k) D= ( Y Yˆ ) = eˆ i S = ( Y Y) YY i i i 2 2 i i i i 2

20 The regression equation is cons = income wealth Predictor Coef Stdev t-ratio p Constant income wealth R-sq = Analysis of Variance (ANOVA table) SOURCE DF SS MS F p Regression < Error Total F = 92.4, p-value in F 2,7 is <0.0001

21 Analysis of Variance (ANOVA table) SOURCE DF SS MS F Regression k-1 S YY -D (S YY -D)/(k-1) (S YY -D)/(k-1) 2 error T-k D = e i D/(T-k) D/(T-k) Total T-1 S YY Analysis of Variance (ANOVA table) SOURCE DF SS MS F p Regression < Error Total

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23 Interpreting the F test since the p-value for F = 92.4 is < 0.05 the H o is reject this means H : b = b =...= b = O is rejected 1 2 k 1 0 indicating that one or more b i is unequal to zero 35

24 Multicollinearity b -hence t- for any one variable will change as X variable(s) added or subtracted EXCEPT when the X variables not correlated with each other in which case t is the same in bivariate and multiple regression 37

25 Multicollinearity : consumption, income and wealth example A) The regression equation is consumption = income Predictor Coef Stdev t-ratio p Constant income B) The regression equation is consumption = income wealth Predictor Coef Stdev t-ratio p Constant income wealth Correlation of income and wealth = 0.999

26 Multicollinearity in A) income is highly significant similarly wealth alone is highly significant in B) wealth and income are insignificant yet paradoxically R 2 = 0.96 the reason for such extreme changes in the apparent effects is multicollinearity 39

27 Multicollinearity multicollinearity means that the X variables are very highly correlated, so that they are not distinct so with severe multicollinearity the estimated b s become very unreliable if we change X 1 (say) slightly, b 1 changes a lot the standard errors of the b s become very large hence we see low t values but high R 2 40

28 Multicollinearity Ln Halifax House price index, Greater London and Scotland Model 1: OLS, using observations 1983:2-2007:2 (T = 97) Dependent variable: lns coefficient std. error t-ratio p-value const e-017 *** lngl e-037 *** Mean dependent var S.D. dependent var Sum squared resid S.E. of regression R-squared Adjusted R-squared F(1, 95) P-value(F) 7.26e-37 Model 2: OLS, using observations 1983:2-2007:2 (T = 97) Dependent variable: lns coefficient std. error t-ratio p-value const e-019 *** lngl e-07 *** lne_ro e-028 *** Mean dependent var S.D. dependent var Sum squared resid S.E. of regression R-squared Adjusted R-squared F(2, 94) P-value(F) 4.36e-62

29 lns, lne_ro and lngl 7 lns lngl lne_ro

30 Solutions to multicollinearity problems use less-correlated X variables eg data for a longer/different time period, so that X 1, X 2, etc become more separated use the change in Y, X at each point in time rather than the levels of Y, X, since changes tend not to be as strongly correlated as levels 43

31 Solutions to multicollinearity problems use the change in Y, X at each point in time Actually the difference in logs equals the exponential growth rate X(t)=105 X(t-1)=100 growth 5% LnX(t) = lnx(t-1) = lnx(t)-lnx(t-1)=

32 Solutions to multicollinearity problems : use differences = growth with logs Model 3: OLS, using observations 1983:2-2007:2 (T = 97) Dependent variable: d_lns coefficient std. error t-ratio p-value const ** d_lngl d_lne_ro e-06 *** Mean dependent var S.D. dependent var Sum squared resid S.E. of regression R-squared Adjusted R-squared F(2, 94) P-value(F) 1.00e-06 45

33 Fitted values versus Scotland house price growth fv d_lns

34 Fitted values versus Scotland house price growth : with quarterly dummies Model 4: OLS, using observations 1983:2-2007:2 (T = 97) Dependent variable: d_lns coefficient std. error t-ratio p-value const * d_lngl ** d_lne_ro e-05 *** dq * dq *** dq Mean dependent var S.D. dependent var Sum squared resid S.E. of regression R-squared Adjusted R-squared F(5, 91) P-value(F) 2.70e-09 48

35 Observed S growth and fitted values : with quarterly dummies fv d_lns

36 F test of seasonal effects Model 5: OLS, using observations 1983:2-2007:2 (T = 97) Dependent variable: d_lns coefficient std. error t-ratio p-value const ** d_lngl d_lne_ro e-06 *** Mean dependent var S.D. dependent var Sum squared resid S.E. of regression R-squared Adjusted R-squared F(2, 94) P-value(F) 1.00e-06 Log-likelihood Akaike criterion Schwarz criterion Hannan-Quinn rho Durbin-Watson Comparison of Model 5 and Model 4: Null hypothesis: the regression parameters are zero for the variables dq1, dq2, dq3 Test statistic: F(3, 91) = , with p-value = e

37 Summary Statistical criteria R 2 gives the overall % of Y s variation accounted for by the X variables t test for testing the significance of individual Xs F test for testing whether groups of Xs should be present in the model Multicollinearity is a problem that occurs when we have highly correlated variables (as often occurs in time series) Solve by reducing the correlation by differencing and/or extra data 52