WORKSHOP. Introductory Econometrics with EViews. Asst. Prof. Dr. Kemal Bağzıbağlı Department of Economic

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1 WORKSHOP on Introductory Econometrics with EViews Asst. Prof. Dr. Kemal Bağzıbağlı Department of Economic Res. Asst. Pejman Bahramian PhD Candidate, Department of Economic Res. Asst. Gizem Uzuner MSc Student, Department of Economic

2 EViews Workshop Series Agenda 1. Introductory Econometrics with EViews 2. Advanced Time Series Econometrics with EViews a. b. c. d. e. 3. Unit root test and cointegration Vector Autoregressive (VAR) models Structural Vector Autoregressive (SVAR) models Vector Error Correction Models(VECM) Autoregressive Distributed Lag processes Forecasting, and Volatility Models with EViews a. b. c. Forecasting Volatility models Regime Switching Models 2

3 Part 1 - Outline 1. Violation of Classical Linear Multiple Regression (CLMR) Assumptions a. b. Heteroskedasticity Multicollinearity c. d. Model Misspecification Autocorrelation 2. Stationarity is Job 1! 3. Univariate Time Series Modelling a. Autoregressive Integrated Moving Average (ARIMA) model 3

4 1. Violation of Classical Linear Multiple Regression (CLMR) Assumptions

5 Multiple Regression Model Deterministic components Stochastic component n observations on y and x: α & βi: unknown parameters 5

6 Assumptions 1) The error term (ut) is a random variable with E(ut )=0. 2) Common (constant) Variance. Var(ut ) = σ2 for all i. 3) Independence of ut and uj for all t. 4) Independence of xj ut and xj are independent for all t and j. 5) Normality ut are normally distributed for all t. In conjunction with assumptions 1, 2 and 3; ut IN (0, σ2) 6

7 Violation of Basic Model Assumptions HETEROSKEDASTICITY (nonconstant variance) var(ut ) = E(ut2) = σ2 for all t (similar distribution) Homoskedasticity: σ12 = σ22 = = σ2n Constant dispersion of the error terms around their mean zero 7

8 Heteroskedasticity (cont.) Rapidly increasing or decreasing dispersion heteroskedasticity? Variances are different because of changing dispersion σ12 σ22... σ2n σt2 Var(ut )= One of the assumptions is violated! 8

9 Heteroskedasticity (cont.) Residuals increasing by x heteroskedasticity? 9

10 Consequences of Heteroskedasticity The ordinary least squares (OLS) estimators are still unbiased but inefficient. Inefficiency: It is possible to find an alternative unbiased linear estimator that has a lower variance than the OLS estimator. 10

11 Consequences of Heteroskedasticity (cont.) Effect on the Tests of Hypotheses The estimated variances and covariances of the OLS estimators are biased and inconsistent invalidating the tests of hypotheses (significance) Effect on Forecasting Forecasts based on the estimators will be unbiased Estimators are inefficient forecasts will also be inefficient 11

12 Lagrange Multiplier (LM) Tests for Heteroskedasticity 1. Park Test is a two-stage procedure Stage 1: Run an OLS regression disregarding the heteroskedasticity question. Obtain ut from this regression; Stage 2: if β is statistically significant, there is heteroskedasticity. 12

13 Park Test in EViews ls compensation c productivity 13

14 Park Test in EViews (cont.) 14

15 Park Test in EViews (cont.) u=0 15

16 Park Test in EViews (cont.) u2=u^2 16

17 Park Test in EViews (cont.) lnu2=log(u2) lnproductivity=log(productivity) 17

18 Park Test in EViews (cont.) Probability value (p-value) of lnproductivity (0.5257) is greater than the critical value of 0.05 Statistically insignificant homoskedasticity 18

19 Detection of Heteroskedasticity (cont.) 2. Glejser Test is similar in spirit to the Park test. Glejser (1969) suggested estimating regressions of the type; IûtI = α + βxt IûtI = α + β/xt IûtI = α + β Xt and so on Testing the hypothesis β=0 19

20 Glejser Test in EViews genr 20

21 Glejser Test in EViews (cont.) ls au c productivity Heteroskedasticity? 21

22 Glejser Test in EViews (cont.) ls au c 1/productivity ls au 22

23 Glejser Test in EViews (cont.) ls compensation c productivity 23

24 Glejser Test in EViews (cont.) 24

25 Detection of Heteroskedasticity (cont.) 3. White s Test Recommended over all the previous tests Step 1: Obtain by OLS Step 2: Compute the residual and square it 25

26 Detection of Heteroskedasticity (cont.) 3. White s Test (cont.) Step 3: Regress the squared residual against a constant, X2t, X3t etc. (auxiliary equation) Step 4: Compute the statistic nr2 n: sample size, R2: unadjusted R2 from S.3 26

27 Detection of Heteroskedasticity (cont.) 3. White s Test (cont.) Step 5: Reject the null hypothesis that if Upper a percent point on the chi-square dist. with 5 d.f. If the null hypothesis is not rejected the residuals are homoskedastic 27

28 White Test in EViews 28

29 Solutions to the Heteroskedasticity Problem Taking the logarithm of Yt and Xt variance becomes smaller. Use the weighted least squares (WLS) Better than the first solution Guaranties homoskedasticity. 29

30 Solutions to the Heteroskedasticity Problem (cont.) Graphical Method Check the residuals (i.e. error variance) linearly increasing with xt WLS 30

31 Solutions to the Heteroskedasticity Problem (cont.) Not linearly but quadratically increasing error variance 31

32 Solutions to the Heteroskedasticity Problem (cont.) Error variance decreasing linearly 32

33 Applications with EViews ls foodexp c totalexp foodexp c totalexp 01.makeresid u 33

34 Applications with EViews (cont.) Command: scat totalexp u heteroskedasticity? 34

35 Applications with EViews (cont.) 35

36 Applications with EViews (cont.) lnfoodexp=log(foodexp) lntotalexp=log(totalexp) 36

37 Applications with EViews (cont.) Command: ls lnfoodexp c lntotalexp 37

38 Applications with EViews (cont.) 38

39 Multicollinearity OLS Ordinary Least Squares BLUE classical normal linear Independent variables in the regression model are not correlated. 39

40 What is Multicollinearity? The problem of multicollinearity arises when the explanatory variables have approximate linear relationships. i.e. explanatory variables move closely together In this situation, it would be difficult to isolate the partial effect of a single variable. WHY? 40

41 Multicollinearity (cont.) 1. Exact (or Perfect) Multicollinearity a. Linear relationship among the independent variables 2. Near Multicollinearity a. Explanatory variables are approximately linearly related For example; If Exact Near 41

42 Theoretical Consequences of Multicollinearity Unbiasedness & Forecasts OLS estimators are still BLUE and MLE and hence are unbiased, efficient and consistent. Forecasts are still unbiased and confidence intervals are valid Although the standard errors and t-statistics of regression coefficients are numerically affected, tests based on them are still valid 42

43 Theoretical Consequences of Multicollinearity (cont.) Standard Errors Standard errors tend to be higher making t-statistics lower thus making coefficients less significant (and possibly even insignificant) 43

44 Identifying Multicollinearity High R2 with low values for t-statistics High values for correlation coefficients Regression coefficients sensitive to specification Formal test for multicollinearity Eigenvalues and condition index (CI) k= max eigenvalues/min eigenvalues CI= k k is between 100 and 1000 multicollinearity? High variance inflation factor (VIF) VIF>10 THEN multicollinearity is suspected. 44

45 Solutions to the Multicollinearity Problem Benign Neglect Less interested in interpreting individual coefficients but more interested in forecasting Eliminating Variables The surest way to eliminate or reduce the effects of multicollinearity 45

46 Solutions to the Multicol. Problem (cont.) Reformulating the Model In many situations, respecifying the model can reduce multicollinearity 46

47 Solutions to the Multicol. Problem (cont.) Using Extraneous Information Often used in the estimation of demand functions High correlation between time series data on real income and the price level Making the estimation of income and price elasticities difficult Estimate the income elasticity from cross-section studies and then use that information in the time series model to estimate the price elasticity 47

48 Solutions to the Multicol. Problem (cont.) Increasing the Sample Size reduces the adverse effects of multicollinearity R2, including the new sample goes down or remains approx. the same the variances of the coefficients will indeed decrease and counteract the effects of multicollinearity goes up substantially there may be no benefit to adding to the sample size 48

49 Applications with EViews Overall statistically significant but one by one statistically insignificant multicollinearity problem 49

50 Applications with EViews (cont.) Command: eq01.varinf 50

51 Applications with EViews (cont.) Command: scalar CI: Condition Index 51

52 Applications with EViews (cont.) The highest correlation is between the price of cars and the general price level. Even if we drop these variables one-by-one from the model, still we have a multicollinearity problem. 52

53 Applications with EViews (cont.) When we drop both the general price level and the price of cars, the multicollinearity problem is solved but R2 is low. So we check the second highest correlation between disposable income and price level. 53

54 Applications with EViews (cont.) DROP: General price level and disposable income After removing the variables, the problem is solved. Loss of valuable information? It is better to try solving the problem by increasing the sample size 54

55 Model Misspecification 1. Omitting Influential or Including NonInfluential Explanatory Variables 2. Various Functional Forms 3. Measurement Errors 4. Tests for Misspecification 5. Approaches in Choosing an Appropriate Model 55

56 The Ramsey RESET Test RESET: Regression specification error test Step 1: Estimate the model that you think is correct and obtain the fitted values of Y, call them Step 2: Estimate the model in Step 1 again, this time include as additional explanatory variables. 56

57 The Ramsey RESET Test (cont.) Step 3: The model in Step 1 is the restricted model and the model in Step 2 is the unrestricted model. Calculate the F-statistic for these two models. i.e. carry out a Wald F-test for the omission of the two new variables in Step 2 If the null hypothesis (H0: the new variables have no effect) is rejected indication of a specification error 57

58 Autocorrelation In the presence of autocorrelation, cov( ut,us ) 0 for t s and the error for period t is correlated with the error for period s. -1< ρ <1 ρ approaching 0 ρ approaching +1 ρ approaching -1 no correlation positive correlation negative correlation 58

59 Autocorrelation (cont.) 59

60 Causes of Autocorrelation DIRECT INDIRECT Inertia or Persistence Omitted Variables Spatial Correlation Functional Form Cyclical Influences Seasonality 60

61 Consequences of Autocorrelation OLS estimates are still unbiased and consistent OLS estimates are inefficient not BLUE Forecasts will also be inefficient The same as the case of ignoring heteroskedasticity Usual formulas give incorrect standard errors for OLS estimates Confidence intervals and hypothesis tests based on the usual standard errors are not valid 61

62 Detecting Autocorrelation Runs Test: Investigate the signs of the residuals. Are they moving randomly? (+) and (-) comes randomly don t need to suspect autocorrelation problem. Durbin-Watson (DW) d Test: Ratio of the sum of squared differences in successive residuals to the residual sum of squares. Breusch-Godfrey LM Test: A more general test which does not assume the disturbance to be AR(1). 62

63 Durbin-Watson d Test STEP 1 Estimate the model by OLS and compute the residuals ut STEP 2 Compute the Durbin-Watson d statistic: 63

64 Durbin-Watson d Test (cont.) STEP 3 Construct the table with the calculated DW statistic and the du, dl, 4-dU and 4-dL critical values. STEP 4 Conclude 64

65 Resolving Autocorrelation The Cochrane-Orcutt Iterative Procedure Step 1: Estimate the regression and obtain residuals. Step 2: Estimate the first-order serial correlation coefficient ( ) from regressing the residuals to its lagged terms. Step 3: Transform the original variables as follows: 65

66 Resolving Autocorrelation (cont.) Step 4: Run the regression again with the transformed variables and obtain a new set of residuals. Step 5 and on: Continue repeating Steps 2 to 4 for several rounds until the following stopping rule applies: the estimates of from two successive iterations differ by no more than some preselected small value, such as

67 Applications with EViews Variables in natural logarith: LNCO: Copper price LNIN: Inudtrial production LNLON: London stock exchange LNHS: Housing price LNAL: Aluminium price AUTOCORRELATION? 67

68 Applications with EViews (cont.) H0: No autocorrelation 68

69 Applications with EViews (cont.) To Fix it! 69

70 Applications with EViews (cont.) To Fix it! u=u(0) 70

71 Applications with EViews (cont.) To Fix it! Generate series: y= lnco-0.52*lnco(-1) x2= lnin-0.52*lnin(-1) x3= lnlon-0.52*lnlon(-1) x4= lnhs-0.52*lnhs(-1) x5= lnal-0.52*lnal(-1) 71

72 Applications with EViews (cont.) To Fix it! Command: ls y c x2 x3 x4 x5 72

73 Applications with EViews (cont.) To Fix it! 73

74 Summary Problem Source Detection Remedy Heteroskedasticity Nonconstant variance Park Test, Glejser, White Test Taking logarithm, Weighted least squares Autocorrelation E(ut,ut-1) 0 Durbin-Watson d Test, Run Test, Breusch Godfrey LM Test Cochrane-Orcutt Iterative Procedure and GLS Multicollinearity Interdependence of xj High R2 but few significant t ratios High pairwise correlation between independent variables Eigenvalues and condition index, High VIF, Auxiliary Regressions Reformulating the model Dropping variables, Additional new data Faitor analysis Principal comp. analysis 74

75 2. Stationarity is Job 1!

76 What is Stationarity? A stationary series can be defined as one with a constant mean, constant variance and constant autocovariances for each given lag. The mean and/or variance of nonstationary series are time dependent. The correlation between a series and its lagged values depend only on the length of the lag and not on when the series started. A series that is integrated of order zero, i.e. I(0). 76

77 Example of a white noise process Time series plot of a random walk vs. a random walk with drift 77

78 Example PDI: Personal Disposable Income 78

79 What is Stationarity? (cont.) If a regression model is not stationary, the usual t-ratios will not follow a t-distribution. The use of nonstationary data can lead to spurious regressions. Results of the regression do not reflect the real relationship except if these variables are cointegrated. 79

80 3. Univariate Time Series Modelling

81 Some Stochastic Processes Random Walk Moving Average Process Autoregressive Process Autoregressive Moving Average Process 81

82 Autoregressive Integrated MA Process Most time series are nonstationary Successive differencing stationarity : A stationary series that can be expressed by an ARMA(p, q) can be represented by an ARIMA model ARIMA(p, d, q) 82

83 Estimation and Forecasting with an ARIMA Model The Box and Jenkins (1970) Approach Identification Fitting (Estimation), usually OLS Diagnostics Refitting if necessary Forecasting 83

84 Identification The process of specifying the orders of differencing, AR modeling, and MA modeling How do the data look like? What pattern do the data show? - Are the data stationary? - Specification of p, d, and q? Tools - Plots of data - Autocorrelation Function (ACF) - Partial ACF (PACF) 84

85 Identification (cont.) To determine the value of p and q we use the graphical properties of the autocorrelation function and the partial autocorrelation function. Again recall the following: 85

86 Model Fitting Model parameters are estimated by OLS Output includes Parameter estimates Test statistics Goodness of fit measures Residuals Diagnostics 86

87 Diagnostics Determines whether the model fits the data adequately. The aim is to extract all information and ensure that residuals are white noise Key measures ACF of residuals PACF of residuals Ljung-Box Pierce Q statistic 87

88 Preliminary Analysis with EViews Select the series dividends in the workfile, then select [Quick/Graph/Line graph]: 88

89 Preliminary Analysis with EViews (cont.) [Quick/Generate Series]: ddividends=d(dividends) 89

90 Preliminary Analysis: Identification Correlogram The graph of autocorrelation function against s, for s = 0, 1, 2,, t-1 Useful diagram for identifying patterns in correlation among series. Useful guide for determining how correlated the error term (ut ) is to the past errors ut-1, ut-2,... 90

91 Preliminary Analysis: Identification Interpretation of Correlogram If is high, correlogram for AR (1) declines slowly over time First differencing is indicated 91

92 Preliminary Analysis: Identification Interpretation of Correlogram The function quickly decreases to zero (a low ) 92

93 Correlogram and Stationarity 93

94 Preliminary Analysis: Estimation ARIMA(1,1,1) Command: ls ddividens c AR(1) MA(1) 94

95 Empirical Example Forecasting Monthly Electricity Sales Total System Energy Demand 95

96 Empirical Example (cont.) Forecasting Monthly Electricity Sales Correlogram for Monthly Electricity Sales Data 96

97 Empirical Example (cont.) Forecasting Monthly Electricity Sales Correlogram for 12-Month Differenced Data (Xt-Xt-12) 97

98 Empirical Example (cont.) Forecasting Monthly Electricity Sales ARMA Order AIC RMSE (1, 1) 1, (4, 1) 1, (1, 4) 1, (0, 4) 1, Box-Jenkins Forecast of System Energy RMSE: Root mean squared error Superior model: ARIMA (0, 1, 4) 98

99 Bibliography Brooks, C. (2008) Introductory Econometrics for Finance, Gujarati D.N., Porter D.C. (2004), Basic Econometrics,The McGraw Hill Companies Maddala, G.S. (2002). Introduction to Econometrics. Ramanathan, R. (2002). Introductory econometrics with applications, Thomson Learning. Mason, Ohio, USA. Wooldridge,J. (2000) Introductory Econometrics: A modern Approach. South-Western College Publishing 99