11/18/2008. So run regression in first differences to examine association. 18 November November November 2008

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1 Time Series Econometrics 7 Vijayamohanan Pillai N Unit Root Tests Vijayamohan: CDS M Phil: Time Series 7 1 Vijayamohan: CDS M Phil: Time Series 7 2 R 2 > DW Spurious/Nonsense Regression. Integrated but mutually independent time series; (high serial correlation in each series high correlation); Integrated series: Unit root So run regression in first differences to examine association. Further empirical evidences for first differencing in regression: Plosser and Schwert (1978). Significance of differencing: in ARIMA modelling. Question of choice between differencing and detrending Recognizing the differentiation between TSP and DSP. TSP: stationary around a deterministic trend: DSP: Only stochastic trend: x t = + t ; y = t 1 : as is stationary. Discrimination between TSP and DSP models: essential; whether the root of the series = 1 or 1. Hence the significance of unit root tests. Vijayamohan: CDS M Phil: Time Series 7 3 Vijayamohan: CDS M Phil: Time Series 7 4 1

2 Consider the following model: = + t t We have the following possibilities: 1. When 0 1 U t : white noise. has a linear trend and hence is a TSP. 2. When = 0 then = Given = + t t When = 0 = We have two cases: (i) 1 is a stationary series; (ii) = 1 is a DSP with a drift. 3.When = = 0 then = -1. Vijayamohan: CDS M Phil: Time Series 7 5 Vijayamohan: CDS M Phil: Time Series 7 6 Given = + t t When = = 0 = -1 Two cases here are: (i) 1 is stationary; (ii) = 1 is a DSP without drift. Rewrite (1) = + t t = + t t u t Now testing the null hypothesis Ho: = 0 as where = ( 1). in the usual way is equivalent to testing Ho: = 1. Vijayamohan: CDS M Phil: Time Series 7 7 Vijayamohan: CDS M Phil: Time Series 7 8 2

3 ly rewrite (2) = + -1 Vijayamohan: CDS M Phil: Time Series 7 9 as = and (3) = -1 as = -1. Then test for Ho: = 0 vs. the one-sided alternative 1. Dickey-Fuller (DF) Unit Root Test: 3 DF Type Formulations Note: In (1) = + t t u t where = ( 1) the model has both a constant and a trend; Note: In (2) = + -1 the model has a constant; Note: In (3) = -1 the model is without constant. Vijayamohan: CDS M Phil: Time Series 7 10 and In a DF test all 3 formulations are considered. But we cannot use the usual t-test to test Ho: = 1 because under the null is I(1) and hence the t-statistic does not have an asymptotic normal distribution (Dickey and Fuller 1979). Its asymptotic distribution based on Wiener processes is called Dickey-Fuller distribution and the statistic Dickey-Fuller (tau) statistic. Critical values tabulated by Dickey and Fuller (1979) and MacKinnon (1990) for a wider range of sample. Most of the statistic outcomes are negative; Note: Ho: = ( 1) = 0. If the estimated value is more negative (i.e. less) than the critical value at the chosen significance level reject Ho. Vijayamohan: CDS M Phil: Time Series 7 11 Vijayamohan: CDS M Phil: Time Series

4 Augmented Dickey-Fuller (ADF) Unit Root Test In deriving the asymptotic distributions Dickey and Fuller ( ) assumed: u t iid(0 2 ). But if the errors are non-orthogonal (i.e. serially correlated) the limiting distributions cease to be appropriate. Vijayamohan: CDS M Phil: Time Series 7 13 Augmented Dickey-Fuller (ADF) Unit Root Test Dickey and Fuller (1979) and Said and Dickey (1984) modified the DF test by means of AR correction: Augmented Dickey-Fuller test (ADF): by estimating an autoregression of on its own lags and -1 using OLS: y y When = 0 = 1. t The (t -) test statistic follows the same DF distribution ( statistic) Vijayamohan: CDS M Phil: Time Series 7 14 p t 1 i y t 1 ut. i 1 Test for a unit root in Y t ; If the unit root null is not rejected (if Y t appears I(1)) test for a second unit root (see if Y t is I(2)): Testing: (1): Double Unit Roots Testing Estimate the regression of 2 Y t on a constant Y t 1 and the lagged values of 2 Y t and compare the t-ratio of the coefficient of Y t 1 with the DF critical values. Testing: (2): Double Unit Roots Testing Estimate the regression of 2 Y t on Y t 1 Y t 1 and the lagged values of 2 Y t and compute the usual F-statistic for testing the joint significance of Y t 1 and Y t 1 using the critical values given as 1 (2) by Hasza and Fuller (1979). Vijayamohan: CDS M Phil: Time Series 7 15 Vijayamohan: CDS M Phil: Time Series

5 Testing for Unit Root: (Augmented) Dickey - Fuller test: Consumption Testing for Unit Root: (Augmented) Dickey - Fuller test: EQ( 1) Modelling C t by OLS (using Data.in7) The estimation sample is: 1953 (2) to 1992 (3) Coefficient Std.Error t-value t-prob Constant C t sigma RSS R^ F(1156) = [0.418] log-likelihood DW 1.6 no. of observations 158 no. of parameters 2 Consumption Vijayamohan: CDS M Phil: Time Series 7 17 C t = C t 1 t = (0.793) ( 0.812) Critical values used in ADF test: 5%= %= Vijayamohan: CDS M Phil: Time Series 7 18 Testing for Unit Root: (Augmented) Dickey - Fuller test: EQ( 1) Modelling 2 C t by OLS (using Data.in7) The estimation sample is: 1953 (3) to 1992 (3) Coefficient Std.Error t-value t-prob Constant C t sigma RSS R^ F(1155) = [0.000]** log-likelihood DW 2.04 no. of observations 157 no. of parameters 2 2 C t = C t 1 t = ( 0.857) ( ) Vijayamohan: CDS M Phil: Time Series 7 19 Critical values used in ADF test: 5%= %= Testing for Unit Root: (Augmented) Dickey - Fuller test: Variable C t in level CONS: ADF tests (T=152 Constant; 5% = % = -3.47) D-lag t-adf beta Y_1 sigma t-dy_lag Vijayamohan: CDS M Phil: Time Series 7 20 t-prob AIC F-prob

6 Testing for Unit Root: (Augmented) Dickey - Fuller test: Variable C t in First Difference DCONS: ADF tests (T=152 Constant; 5% = % = -3.47) D-lag t-adf beta Y_1 sigma t-dy_lag t-prob AIC ** F-prob ** ** ** ** ** Unit Root Test for TSP vs. DSP: Nelson and Plosser (1982) The first ever attempt: Nelson and Plosser (1982) using the (augmented) Dickey-Fuller unit root tests: H 0 : a time series belongs to DSP class against H a : it belongs to TSP class. Nelson and Plosser found that 13 out of 14 US macroeconomic time series that they analyzed belonged to the DSP class (the exception being the unemployment rate). Vijayamohan: CDS M Phil: Time Series 7 21 Vijayamohan: CDS M Phil: Time Series 7 22 Series Results of Nelson and Plosser (1982 Table 5 p. 151): Critical value at 5% level: 3.45 Sample size (T) Lag (k) -value Real GDP Nominal GDP Real per capita GNP Industrial production Employment Unemployment rate * GNP deflator Consumer prices Wages Real wages Money stock Velocity Interest rate Common 18 November stock 2008 prices Vijayamohan: 100 CDS M Phil: Time Series Unit Root Test for TSP vs. DSP: Nelson and Plosser (1982) They started with a TSP model in which the errors are serially correlated (in first order): = + t and u t = u t 1 e t is assumed to be Gaussian white noise. Nesting these two models gives: = + t + [ 1 (t 1)] = + t + 1 Vijayamohan: CDS M Phil: Time Series 7 24 where or = (1 ) + and = (1 ). 6

7 Unit Root Test for TSP vs. DSP: Nelson and Plosser (1982) = + t + 1 where = (1 ) + and = (1 ). Note: If = 1 then = 0 and (under the null) is a random walk with drift (i.e. DSP); and under the alternative is a TSP. Unit Root Test for TSP vs. DSP: Nelson and Plosser (1982) This is known as Bhargava type formulation for unit root testing (Bhargava 1986). In a second model with a constant only (i.e. no trend 0 = 0) in the Bhargave type formulation we have = + 1 where = (1 ); Vijayamohan: CDS M Phil: Time Series 7 25 Vijayamohan: CDS M Phil: Time Series 7 26 Unit Root Test for TSP vs. DSP: Nelson and Plosser (1982) = + 1 where = (1 ); Note: if = 1 then = 0 and (under the null) we have a drftless random walk or DS series. Under the alternative 1 is a stationary series around /(1 ). DF Type Formulation vs. Bhargava Type Formulation The Dickey-Fuller Type formulations are: = + t + -1 (1) = + -1 (2) = -1 (3) Problems with these formulation: The parameters in the first two functions have different interpretations under the null and the alternative (Schmidt and Phillips 1992). Vijayamohan: CDS M Phil: Time Series 7 27 Vijayamohan: CDS M Phil: Time Series

8 DF Type Formulation vs. Bhargava Type Formulation In DF Type formulation (1): = + t + -1 under the unit root null and represent coefficients of t and t 2 in a quadratic trend while under the alternative they represent the level and the coefficient of t in a linear trend. In DF Type formulation (2): = + -1 under the null hypothesis represents the coefficient of t in a linear trend whereas under the alternative there is no trend and is stationary around /(1 ). DF Type Formulation vs. Bhargava Type Formulation In Bhargava type formulation no such problems of interpretation: In (1): = + t + 1 where = (1 ) + and = (1 ) If = 1 then = 0 and under the null is a random walk with drift (DSP); and under the alternative 1 is a TSP. Vijayamohan: CDS M Phil: Time Series 7 29 Vijayamohan: CDS M Phil: Time Series 7 30 DF Type Formulation vs. Bhargava Type Formulation In Bhargava type formulation no such problems of interpretation: In (2): = + 1 where = (1 ); If = 1 then = 0 and under the null is a drftless random walk (DSP) and under the alternative is stationary around /(1 ). Vijayamohan: CDS M Phil: Time Series 7 31 Other Unit Root Tests Mushrooming of Unit root tests ever since N and P (1982) feat: 1. Sargan and Bhargava (1983): based on DW statistic; 2. Phillips and Perron (1988): non-parametric test; 3. Cochrane (1988): Variance ratio test; 4. Sims (1988): Bayesian approach to unit root testing; 5. Perron (1989): unit root test under structural break; 6. Pantula and Hall (1991): IV test in ARMA models; 7. Schmidt and Phillips (1992): LM test; Vijayamohan: CDS M Phil: Time Series

9 Other Unit Root Tests 8. Choi (1992): Pseudo-IV estimator test; 9. Yap and Reinsel (1995): Likelihood ratio test in ARMA models; 10. Leybourne (1995): based on forward and reverse DF regressions; 11. Park and Fuller (1995): Weighted symmetric estimator test; 12. Elliott Rothenberg and Stock (1996): Dickey- Fuller GLS test; Tests with stationarity as null: 1. Park (1990) s J-test; Other Unit Root Tests 2. Kwiatkowski Phillips Schmidt and Shin (1992): (KPSS) test; 3. Bierens and Guo (1993): based on Cauchy distribution; 4. Leybourne and McCabe (1994): Modified KPSS test; 5. Choi (1994): based on testing for a MA unit root; 6. Arellano and Pantula (1995): based on testing for a MA unit root. Vijayamohan: CDS M Phil: Time Series 7 33 Vijayamohan: CDS M Phil: Time Series 7 34 Panel data unit root tests: Levin and Lin (1993); Breitung and Meyer (1994) Quah (1994); Other Unit Root Tests Pesaran and Shin (1996); Unit Roots Test: Summing up Why so many unit root tests? There is no uniformly powerful test for the unit root hypothesis (Stock 1994). Why are we interested in testing for unit roots? We need unit root tests as a prelude to cointegration analysis. If so unit root tests are pre-tests; then shouldn t the significance level be much higher (say 25% or more)? Isn't it Data-mining? Vijayamohan: CDS M Phil: Time Series 7 35 Vijayamohan: CDS M Phil: Time Series

10 In summary Unit Roots Test: Summing up it is high time we asked the question: Why all this unit root testing rather than keep suggesting more and more unit root tests and use the Nelson-Plosser data as a guinea pig for every unit root test suggested. Maddala and Kim (1998: 146). Vijayamohan: CDS M Phil: Time Series 7 37 Vijayamohan: CDS M Phil: Time Series