Econometrics. 9) Heteroscedasticity and autocorrelation

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1 30C00200 Econometrics 9) Heteroscedasticity and autocorrelation Timo Kuosmanen Professor, Ph.D.

2 Today s topics Heteroscedasticity Possible causes Testing for heteroscedasticity Modeling heteroscedasticity Autocorrelation Sources AR(1) model of autocorrelated error term Testing for autocorrelation Modeling autocorrelation

3 Assumptions <-> properties Finite sample properties Required assumptions Unbiasedness Exogeneity Efficiency Exogeneity, No autocorrelation, Homoscedasticity Asymptotic properties Consistency Exogeneity, No autocorrelation Asymptotic normality Exogeneity, No autocorrelation, Homoscedasticity

4 What is heteroscedasticity? Recall the statistical assumptions of the linear regression model: Statistical assumption: MLR.5 Homoskedasticity: Var(ε x) = σ 2 If the variance of ε i is not the same across all observations i=1,,n, then the disturbances are said to be heteroscedastic.

5 Causes of heteroscedasticity Can be a symptom of model misspecification Additive model of disturbances may be unrealistic In many applications, disturbances appear to be proportional to y In cross-country data, larger countries tend to have larger residuals (in absolute value) Using the log-linear model that assumes multiplicative disturbances can help: y = f(x) exp(ε) => ln y = lnf(x) + ε Sometimes wrong specification of the functional form may appear as heteroscedasticity For example, fitting a linear model when the true relationship is nonlinear

6 Implications of heteroscedasticity OLS estimator remains unbiased and consistent assumption of homoscedasticity was not needed for proving these properties the problems are not so serious as in the case of endogeneity More efficient estimators may be available This requires some assumptions about heteroscedasticity Conventional statistical inferences fail Standard errors can under- or over-estimate the variance

7 Detecting heteroscedasticity Visual inspection Single regression model: plot the scatter of y and x variables and the regression line Multiple regression: The residuals versus fitted y plot (rvf) Goldfeld-Quandt (1965) test Breusch-Pagan (1979) test White (1980) test General: does not assume particular type of heteroscedasticity May have low power

8 Espoo housing market example: scatter of size (m2) and price ( )

9 Residual e Espoo housing market example (2SLS): Residuals versus fitted (rvf) plot Fitted y

10 Production function of electricity distribution firms: Residuals versus fitted (rvf) plot Residual e Fitted y

11 Breusch-Pagan test in Stata. regress lnenergy lnopex lnkrepl Source SS df MS Number of obs = 160 F( 2, 157) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = lnenergy Coef. Std. Err. t P> t [95% Conf. Interval] lnopex lnkrepl _cons estat hettest Breusch-Pagan / Cook-Weisberg test for heteroskedasticity Ho: Constant variance Variables: fitted values of lnenergy chi2(1) = 2.51 Prob > chi2 =

12 Testing heteroscedasticity Estimate the original model by OLS. Save the residuals e Compute the squared values of the residuals: e i 2 Breusch-Pagan test: Estimate the following linear regression model with OLS: e i2 = β 1 + β 2 ŷ i + u i White test: e i2 = β 1 + β 2 x 2i + β 3 x 3i + + β K x Ki + γ 22 x 2i2 + γ 23 x 2i x 3i + + γ 2K x 2i x Ki + γ 33 x 3i2 + γ 34 x 3i x 4i + + γ 3K x 3i x Ki + γ KK x Ki2 )+ u i The idea is to explain the squared residuals by the original regressors (+ their squared values and cross-products)

13 Testing heteroscedasticity Breusch & Pagan test: If a specific model of heteroscedasticity is assumed, then we can specify the regression equation more precisely Saves degrees of freedom the more powerful test White test: Flexible quadratic functional form No prior assumption about heteroscedasticity needed Power decreases as the number of explanatory variables grows Also other explanatory factors z (not included as regressors of the original model) can be used.

14 Testing heteroscedasticity H 0 : Homoscedasticity: Var(ε i ) = σ 2 for all i=1,,n H 1 : Heteroscedasticity: Var(ε i ) Var(ε j ) for some i,j Test statistics: LR = nr 2 Where R 2 is the coefficient of determination of the regression of e 2 If H 0 holds, then the test statistic follows chi-squared distribution (χ 2 ). The degrees of freedom is equal to the number of regressors used for explaining e 2 (excluding constant). =CHIINV(prob.;df)

15 How to deal with heteroscedasticity? A) Keep using the OLS estimator, but replace the standard errors by White s (1980) heteroscedasticity consistent robust standard errors OLS remains unbiased and consistent Improving efficiency requires further assumptions Note: in Stata, select the option robust to obtain the heteroscedasticity consistent standard errors B) Assume some model of heteroscedasticity that allows you to estimate Var(ε i ). Given the estimated variances, use the generalized least squares estimator (Aitken 1935)

16 CD production function of electricity distribution firms, robust standard errors. regress lnenergy lnopex lnkrepl, vce(robust) Linear regression Number of obs = 160 F( 2, 157) = Prob > F = R-squared = Root MSE = Robust lnenergy Coef. Std. Err. t P> t [95% Conf. Interval] lnopex lnkrepl _cons regress lnenergy lnopex lnkrepl Source SS df MS Number of obs = 160 F( 2, 157) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = lnenergy Coef. Std. Err. t P> t [95% Conf. Interval] lnopex lnkrepl _cons

17 Generalized least squares Suppose Var(ε i ) = σ i 2 are known. Consider the weighted least squares (WLS) estimator: min st.. n i 1 2 i 2 i y x... x i 1 2 2i K Ki i Intuitively, the WLS estimator assigns a higher weight to observations with lower σ i 2

18 Generalized least squares Alternatively, we could transform the data by dividing each observation with σ i, and apply the usual OLS estimator (without intercept) to the transformed data min st.. n i 1 u 2 i y 1 x x... u i 2i Ki 1 2 K i i i i This OLS estimator is equivalent to the previous WLS estimator (in econometrics, this is referred to as generalized least squares) i Aitken (1935) has shown that the GLS estimator is efficient (generalizing the Gauss-Markov theorem)

19 Feasible generalized least squares To make the GLS estimator feasible, we need to estimate the variances σ i2 in one way or another. Note: it is hopeless to try estimate n different variance parameters (σ 12, σ 22,, σ n2 ) from the data of n observations. Some model of heteroscedasticity is needed! A textbook example is to assume σ i to be directly proportional to one of explanatory variables x. Alternatively, we could use the fitted values from a regression lne i2 = β 1 + β 2 x 2 + β 3 x 3 + β K x K + u i as estimates of σ i

20 How time series and cross sections differ? Cross section i =1,,n represents a randomly drawn sample from the population Time series t = 1,,T describes a (random) path of a variable in a time window [1, T] Time series have a natural chronological order, whereas in a cross section the ordering of observations does not matter Path dependence: correlations across time periods (t, t+1) Autocorrelation (serial correlation) 20

21 Macroeconomic data: GDP Inflation rate Examples of time series Financial market data: Asset returns Stock prices Management Science: Big data (new types of data from internet, social media, etc.): Google Trends Facebook likes 21

22 30E00800 Time Series Analysis (6 cr) Teaching Period: IV-V Spring Topics: regression analysis of time series, ARMA models, forecasting, cointegration, ARCH and GARCH models, panel data. Lectures 40 h, Tomi Seppälä Grading: final exam (50%), exercises (20%) and project work (30%). Literature: Brooks, Chris: Introductory econometrics for finance, 2nd edition,

23 Static vs. Dynamic time series models Static model y t = β 1 + β 2 x t + ε t, t=1,,t. Allow for autocorrelation: Cov[ε t, ε s ] 0 Dynamic models Lagged explanatory variable(s) y t = β 1 + β 2 x t + β 3 x t-1 + ε t, t=1,,t. Lagged dependent variable(s) y t = β 1 + β 2 x t + γy t-1 + ε t, t=1,,t.

24 Example: Jevons sunspot theory of business cycle Motivation: What causes business cycle (booms, recessions)? Economic theory predicts a stable equilibrium. Rationale: Sunspots have cyclical fluctuations -> affect the weather -> affect agricultural production -> affect prices -> economy Empirical evidence? Jevons was among the first economists to try find empirical support for his theory. Econometrics of 1870s. 24

25 Jevons sunspot theory revisited Dependent variable: Inflation rate in Finland (π t ) Calculated as the annual percentage change of the GDP deflator Statistics Finland (Historical national accounts) Period: Independent variables: International sunspot number (x t ) Annual percentage change of the sunspot number (z t ) Source: NASA, Monthly data since year 1749 Models: A) π t = α + βx t + ε t, t = 1861,,2012 B) π t = α + βx t + γz t + ε t, t = 1861,,

26 Sunspot cycle (Data source: NASA) 26

27 Jevons sunspot theory revisited Results OLS regression Model A: var8 = inflation, var9 = sunspot number. regress var8 var9 Source SS df MS Number of obs = 152 F( 1, 150) = 5.86 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE =.1261 var8 Coef. Std. Err. t P> t [95% Conf. Interval] var e _cons Estimated model: π t = x t 27

28 Jevons sunspot theory revisited Results OLS regression Model B: var8 = inflation, var9 = sunspot number, var10 = percentage change of the sunspot number of. regress var8 var9 var10 Source SS df MS Number of obs = 152 F( 2, 149) = 3.44 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = var8 Coef. Std. Err. t P> t [95% Conf. Interval] var e var _cons Estimated model: π t = x t z t 28

29 Statistical properties of OLS in the case of autocorrelated error term Finite sample properties OLS remains unbiased OLS is inefficient Asymptotic properties OLS remains consistent (under certain assumptions; see the next slide) OLS is no longer asymptotically efficient

30 Consistency of OLS under autocorrelation Sufficient condition for the consistency of OLS is that [x t, ε t ] are jointly stationary and ergodic. Intuitively, the correlations between disturbances must die off sufficiently rapidly as the observations become further apart in time For example, for AR(1) disturbances ρ ts = ρ t s. The autocorrelation tends to zero as t s increases. Thus, OLS is consistent even if ε t are autocorrelated following AR(1).

31 Autoregressive (AR) error term First-order autoregressive AR(1) error term ε t = ρε t-1 + u t ρ is autocorrelation coefficient and u t is stationary, nonautocorrelated white noise process with E[u t ]=0 and Var[u t ]=σ u 2

32 Autoregressive (AR) disturbances Second-order autoregression AR(2) process ε t = ρ 1 ε t-1 + ρ 2 ε t-2 + u t P-order autoregression AR(P) process ε t = ρ 1 ε t-1 + ρ 2 ε t ρ P ε t-p + u t

33 AR(1) disturbances Each disturbance ε t embodies the entire past history of u ε t = ρε t-1 + u t = u t + ρu t-1 + ρ 2 u t-2 + ρ 3 u t-3 Hence, Var[ε t ]= σ u2 + ρ 2 σ u 2 + ρ 4 σ u 2 + ρ 6 σ u2 ρ > 0 ρ < 0 => positive autocorrelation => negative autocorrelation. Process is weakly stationary if ρ satisfy ρ < 1

34 AR(1) disturbances Assuming weak stationarity holds, Var[ε t ]= σ u2 /(1- ρ 2 ) = σ ε 2 Cov[ε t, ε t-s ]= ρ s σ u2 /(1- ρ 2 )

35 OLS residuals of Model A t

36 H 0: no autocorrelation; Testing for autocorrelation H 1: disturbances are AR(1) 1) Estimate the static model by OLS. Save residuals e t 2) Estimate the following model by OLS e e v + t t 1 t 3) Use the standard t test for testing significance of r (the estimated autocorrelation coefficient)

37 Example test of autocorrelation OLS regression var11 = OLS residuals of Model A, var11_01 = lagged OLS residuals of Model A. regress var11 var11_01, noconstant Source SS df MS Number of obs = 151 F( 1, 150) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = var11 Coef. Std. Err. t P> t [95% Conf. Interval] var11_ Estimate of ρ: Statistically significant.

38 Dealing with autocorrelation Autocorrelation can be treated similar to heteroscedasticity, using a simple data transformation (Cochrane & Orcutt 1949) If the estimated autocorrelation coefficient r is statistically significant, take the partial differences: y t * = y t - ρy t-1, x t * = x t - ρx t-1, t= 2,,T t= 2,,T The efficient least squares estimator is obtained by estimating the static model using the differenced data: y t * = β 1 + β 2 x t * + ε t, t= 2,,T

39 Cochrane & Orcutt estimator Model A Model A: var8 = inflation, var9 = sunspot number. prais var8 var9, rhotype(regress) corc ssesearch Iteration 1: rho = , criterion = Iteration 2: rho = , criterion = Iteration 3: rho = , criterion = Iteration 4: rho = , criterion = Iteration 5: rho = , criterion = Iteration 6: rho = , criterion = Iteration 7: rho = , criterion = Iteration 8: rho = , criterion = Iteration 9: rho = , criterion = Cochrane-Orcutt AR(1) regression -- SSE search estimates Source SS df MS Number of obs = 151 F( 1, 149) = 3.12 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = var8 Coef. Std. Err. t P> t [95% Conf. Interval] var e _cons rho Durbin-Watson statistic (original) Durbin-Watson statistic (transformed)

40 Cochrane & Orcutt estimator Model B Model B: var8 = inflation, var9 = sunspot number, var10 = percentage change of the sunspot number. prais var8 var9 var10, rhotype(regress) corc ssesearch noconstant Iteration 1: rho = , criterion = Iteration 2: rho = , criterion = Iteration 3: rho = , criterion = Iteration 4: rho = , criterion = Iteration 5: rho = , criterion = Iteration 6: rho = , criterion = Iteration 7: rho = , criterion = Iteration 8: rho = , criterion = Iteration 9: rho = , criterion = Cochrane-Orcutt AR(1) regression -- SSE search estimates Source SS df MS Number of obs = 151 F( 2, 149) = 5.84 Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = var8 Coef. Std. Err. t P> t [95% Conf. Interval] var var rho Durbin-Watson statistic (original) Durbin-Watson statistic (transformed)

41 Next time Wed 22 March Topic: Time series econometrics

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