Econometrics EViews Res. Asst. Gizem Uzunerfbemoodle.emu.edu.tr/pluginfile.php/46888/mod_resourc… · · 2015-06-10Consequences of Autocorrelation OLS estimates are still unbiased

WORKSHOP on

Introductory Econometrics with EViews

Asst. Prof. Dr. Kemal BağzıbağlıDepartment of Economic

Res. Asst. Pejman BahramianPhD Candidate, Department of Economic

Res. Asst. Gizem UzunerMSc Student, Department of Economic

EViews Workshop Series Agenda1. Introductory Econometrics with EViews

2. Advanced Time Series Econometrics with EViewsa. Unit root test and cointegrationb. Vector Autoregressive (VAR) modelsc. Structural Vector Autoregressive (SVAR) modelsd. Vector Error Correction Models(VECM)e. Autoregressive Distributed Lag processes

3. Forecasting, and Volatility Models with EViewsa. Forecastingb. Volatility modelsc. Regime Switching Models

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Part 1 - Outline1. Violation of Classical Linear Multiple Regression

(CLMR) Assumptions

2. “Stationarity is Job 1!”

3. Univariate Time Series Modellinga. Autoregressive Integrated Moving Average (ARIMA) model

a. Heteroskedasticityb. Multicollinearity

c. Model Misspecificationd. Autocorrelation

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1. Violation of Classical Linear Multiple

Regression (CLMR) Assumptions

Multiple Regression Model

● n observations on y and x:● α & βi: unknown parameters

Deterministic components Stochastic component

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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)

Assumptions

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HETEROSKEDASTICITY (nonconstant variance) var(ut ) = E(ut

2) = σ2 for all t (similar distribution) Homoskedasticity:

● σ12 = σ2

2 = … = σ2n

● Constant dispersion of the error terms around their mean zero

Violation of Basic Model Assumptions

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Heteroskedasticity (cont.)● Rapidly increasing or

decreasing dispersion heteroskedasticity?

● Variances are different because of changing dispersion

● σ12 ≠ σ2

2 ≠ ...≠ σ2n Var(ut )=

σt2

● One of the assumptions is violated! 8

Heteroskedasticity (cont.) Residuals increasing by x

heteroskedasticity?

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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.

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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

1. Park Test is a two-stage procedureStage 1: ● Run an OLS regression disregarding the

heteroskedasticity question. ● Obtain ut from this regression;

Stage 2: ● if β is statistically significant, there is heteroskedasticity.

Lagrange Multiplier (LM) Tests for Heteroskedasticity

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Park Test in EViewsls compensation c productivity

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Park Test in EViews (cont.)

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u=0

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

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

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● Probability value (p-value) of lnproductivity (0.5257) is greater than the critical value of 0.05

● Statistically insignificanthomoskedasticity


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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

Detection of Heteroskedasticity (cont.)

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Glejser Test in EViewsgenr au=@abs(u)

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Glejser Test in EViews (cont.)

Heteroskedasticity?

ls au c productivity

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ls au c 1/productivity ls au c @sqrtproductivity

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

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3. White’s Test● Recommended over all the previous tests

Step 1: Obtain by OLSStep 2: Compute the residual and square it


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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


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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


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White Test in EViews

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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.

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Solutions to the Heteroskedasticity Problem (cont.)

Graphical Method

● Check the residuals (i.e.

error variance)○ linearly increasing with xt

● WLS

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● Not linearly but

quadratically increasing error variance

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● Error variance decreasing

linearly

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Applications with EViewsls foodexp c totalexp foodexp c totalexp 01.makeresid u

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Command: scat totalexp u

heteroskedasticity?

Applications with EViews (cont.)

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

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Command: ls lnfoodexp c lntotalexp


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OLS Ordinary Least Squares

BLUE classical normal linear Independent variables in the regression model are not correlated.

Multicollinearity

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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?

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Multicollinearity (cont.)

1. Exact (or Perfect) Multicollinearitya. Linear relationship among the independent variables

2. Near Multicollinearitya. Explanatory variables are approximately linearly

related

For example; If ➡ Exact

➡ Near

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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

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Standard Errors★ Standard errors tend to be higher

○ making t-statistics lower ○ thus making coefficients less significant (and

possibly even insignificant)

Theoretical Consequences of Multicollinearity (cont.)

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● 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.

Identifying Multicollinearity

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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

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Solutions to the Multicol. Problem (cont.)

● Reformulating the Model ○ In many situations, respecifying the model can

reduce multicollinearity

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● 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


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● 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


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Overall statistically significant

but one by one

statistically insignificant

multicollinearityproblem

Applications with EViews

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Command: eq01.varinf


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Command: scalar ci= @sqrt(66795998/3.44E-06)


CI: Condition Index51


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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.


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● 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.

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


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1. Omitting Influential or Including Non-Influential Explanatory Variables

2. Various Functional Forms 3. Measurement Errors4. Tests for Misspecification 5. Approaches in Choosing an Appropriate

Model

Model Misspecification

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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.

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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

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 no correlation○ ρ approaching +1 positive correlation○ ρ approaching -1 negative correlation

Autocorrelation

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Autocorrelation (cont.)

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Causes of Autocorrelation

DIRECT INDIRECT

● Inertia or Persistence ● Omitted Variables

● Spatial Correlation ● Functional Form

● Cyclical Influences ● Seasonality

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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 valid61

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).

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Durbin-Watson d Test

STEP 1 Estimate the model by OLS and compute the residuals ut

STEP 2 Compute the Durbin-Watson d statistic:

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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 Conclude64

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:

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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 0.001.

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AUTOCORRELATION?

Applications with EViews

1.143 1.739

Variables in natural logarith:● LNCO: Copper price● LNIN: Inudtrial production● LNLON: London stock exchange● LNHS: Housing price● LNAL: Aluminium price

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H0: No autocorrelation

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Applications with EViews (cont.)To Fix it!

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u=u(0)

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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)


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Command: ls y c x2 x3 x4 x5


1.124 1.743

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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

Summary

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2. “Stationarity is Job 1!”

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

Example of a white noise

process

Time series plot of a

random walk vs. a random walk with drift

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ExamplePDI: Personal Disposable Income

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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.

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3. Univariate Time Series Modelling

Some Stochastic ProcessesRandom Walk

Moving Average Process

Autoregressive Process

Autoregressive Moving Average Process

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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

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

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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

● 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:

Identification (cont.)

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● Model parameters are estimated by OLS● Output includes

○ Parameter estimates○ Test statistics○ Goodness of fit measures○ Residuals○ Diagnostics

Model Fitting

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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

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Preliminary Analysis with EViewsSelect the series “dividends” in the workfile, then select [Quick/Graph/Line graph]:

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[Quick/Generate Series]:

Preliminary Analysis with EViews (cont.)

ddividends=d(dividends)89

Preliminary Analysis: IdentificationCorrelogram● 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, ...

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Preliminary Analysis: IdentificationInterpretation of Correlogram● If ⍴ is high, correlogram for AR

(1) declines slowly over time○ First differencing is indicated

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Preliminary Analysis: IdentificationInterpretation of Correlogram● The function quickly decreases

to zero (a low ⍴)

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Correlogram and Stationarity

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

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Empirical ExampleForecasting Monthly Electricity Sales

Total System Energy Demand

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Empirical Example (cont.)Forecasting Monthly Electricity Sales

Correlogram for Monthly Electricity Sales Data

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Correlogram for 12-Month Differenced Data(Xt-Xt-12)

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Box-Jenkins Forecast of System Energy

RMSE: Root mean squared error

Superior model: ARIMA (0, 1, 4)

ARMA Order AIC RMSE

(1, 1) 1,930 320

(4, 1) 1,927 312

(1, 4) 1,926 311

(0, 4) 1,924 311

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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

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Documents

Econometrics EViews Res. Asst. Gizem Uzunerfbemoodle.emu.edu.tr/pluginfile.php/46888/mod_resourc… · · 2015-06-10Consequences of Autocorrelation OLS estimates are still unbiased