Applied Econometrics - Formal Testing of Non-Stationarity vs. Stationarity - Lecture Handout 8

Applied Econometrics Formal Testing of NonStationarity vs. Stationarity Lecture handout 8 Autumn 2023 D.Sc. (Econ.) Elias Oikarinen Professor (Associate) of Economics University of Oulu, Oulu Business School 1 This Handout • Recap of stationary vs. non-stationary time series • How to test for the...

Applied Econometrics Formal Testing of NonStationarity vs. Stationarity Lecture handout 8 Autumn 2023 D.Sc. (Econ.) Elias Oikarinen Professor (Associate) of Economics University of Oulu, Oulu Business School 1 This Handout • Recap of stationary vs. non-stationary time series • How to test for the presence of unit root: Formal tests of non-stationarity vs. stationarity • POLL: 1) Do you know what stationarity is about; 2) Do you readily know, what the unit root means and how to formally test for stationarity? • Brooks, Chapter 8, pp. 353-373 • Enders, pp. 195-243 2 Recap: Stationary vs. Non-Stationary Time Series  Stationary time series has a constant (long-term, i.e. unconditional) variance and mean: Shocks in stationary time series are temporary, i.e., over time the influence of a shock disappears and the time series gets back to its long-term mean value – shocks to the system will gradually die away  Mean and variance of non-stationary time series have permanent components  No long-term mean towards which the series would get back to  Variance has no finite value, when t    Sample autocorrelations die out very slowly  For us, stationarity refers to covariance stationarity (”weak” stationarity) 3 Stationary vs. Non-Stationary Time Series • • • • • • • If regression is run with non-stationary time series: There is a threat of spurious regression (nonsense regression) The usual t-statistic will not follow the t-distribution, and the F-statistic will not follow the F-distribution, and so on  usual inference and testing procedures do not apply Possible estimation of a cointegrating relationship, i.e. a long-run (equilibrium) relationship among two or more non-stationary variables Therefore, time series analysis should always begin by verifying whether the time series in question are stationary or non-stationary Time series are often non-stationary in levels, e.g.: consumer price index, GDP, share prices, housing prices Differenced variables usually are stationary, e.g.: GDP growth, stock market returns, housing price change… 4 Stationary vs. Nonstationary Time Series • Neste total return index (which curve is in natural log form?) 5 90 4,5 80 4 70 3,5 60 3 50 2,5 40 2 30 1,5 1 20 0,5 10 18.4.2020 18.4.2019 18.4.2018 18.4.2017 18.4.2016 18.4.2015 18.4.2014 18.4.2013 18.4.2012 18.4.2011 18.4.2010 18.4.2009 18.4.2008 18.4.2007 18.4.2006 0 18.4.2005 0 5 Daily return 20-day rolling volatility (sd.) 19.4.2020 19.4.2019 19.4.2018 19.4.2017 19.4.2016 19.4.2015 19.4.2014 19.4.2013 19.4.2012 19.4.2011 19.4.2010 19.4.2009 19.4.2008 19.4.2007 19.4.2006 19.4.2005 Stationary vs. Nonstationary Time Series • Neste share return 0,25 0,2 0,15 0,1 0,05 0 -0,05 -0,1 -0,15 6 Spurious Regression: Illustration Based on Simulated Models (Enders, p. 197) Spurious Correlation Examples: Which one is your favorite? 8 Spurious Correlation Example I 9 Spurious Correlation Example II 10 Spurious Correlation Example III 11 Spurious Correlation Example IV 12 Spurious Correlation Example V 13 Unit Root • • • • • If time series has a stochastic trend, it has a unit root Unit root and non-stationarity are not exactly synonymous, though: A stochastic process with a deterministic (but no stochastic) trend is non-stationary but does not have a unit root The concept of unit root can be formally illustrated with the so-called lag operator (e.g. Brooks, pp. 355-356) In practice, the presence of a unit root in yt is typically investigated by testing whether 1 = 1 in the equation: If we conclude that 1 = 1 (or > 1), we conclude that yt has a unit root and therefore is non-stationary 14 Dickey-Fuller Unit Root Test    Dickey-Fuller (DF) test is the most commonly applied formal unit root test (Dickey and Fuller, 1979, 1981) 0-hypothesis:  = 1 in model: In practice, a more practical form of the equation is used in the test: yt-1 is deducted from both sides of the equation to get:  Hence, the 0-hypotheses: 1 = 0 (i.e.  = 1, since 1 =  - 1)  The tested model also can include a constant and a trend:  Conclusion is based on the t-value of parameter 1 15 Unit Root Tests: Deterministic Variables  At least constant should generally be included  If we might suspect that the series is stationary around a deterministic trend (pretty rare), i.e. is trend-stationary: also time trend included  For instance, the rate of return on an asset, or GDP growth rate, cannot really have a deterministic trend  constant but no trend included in the tested model  When testing for model residual stationarity: no deterministic variables  If not sure about what deterministic variables to include and t-value > 0, it makes sense to include constant (or if it is included already, trend too) 16 Dickey-Fuller Unit Root Test   Dickey & Fuller have computed the critical values for each possible three cases: no deterministic variables; constant included; both constant and trend included If t-value of 1 < critical value  null hypothesis of unit root is rejected and it is concluded that the series is stationary (Enders, p. 208:) 17 Augmented Dickey-Fuller Test  Residual from the tested equation should be white noise  A common complications is autocorrelation in t  If t is autocorrelated, the size of the test is too large, i.e., a correct null hypothesis is rejected too often  ADF (Augmented Dickey-Fuller) test provides a solution, where p lags of the dependent variable are included in the tested model:  This is the most often used version of the test  The same parameter (1 ) is under interest with the critical values also being the same as in the DF test 18 Augmented Dickey-Fuller Test: Lag Length   Test result can depend on the lag length If t is autocorrelated, the size of the test is too large, i.e., a correct null hypothesis is rejected too often  If there are numerous lags, power of the test diminishes, i.e., the null hypothesis of unit root is rejected too rarely – in other words, a faulty conclusion that time series is non-stationary is too often made (for a stationary series)  How to choose the lag lenth p?  With monthly data, a good starting point may be 12, with quarterly data 4 19 Augmented Dickey-Fuller Test: Lag Length  In statistical packages such as EViews, it is typically possible to select p based on a variety of different selection criteria  In Eviews, p can be selected based on: SBC (in Eviews it is ”SIC”), AIC, Hannan-Quinn Information Criterion, modified version of the info criteria, and ‘general-to-specific method’ (i.e. “t-test” in Eviews)  Ng and Perron (2001) show that a modified version of the AIC (MAIC) yields a better estimate of the correct lag length than either the AIC or the SBC  MAIC is equal to the usual expression for the AIC plus an additional penalty term for additional lags  See Enders, p. 221 20 Table for (A)DF Test Critical Values Interpretation 1) If the test value < -1.95, we (can rejects the null at 5% sig. level and) conclude that the series is stationary with zero mean. 2) Having 100 obs, if the test value < -3.51, we (can rejects the null at 1% sig. level and) conclude that the series is stationary. 3) With 500 obs and relying on the 10% level of significance, if the test value < -3.13, we conclude that the series is stationary around a deterministic trend. (It might be a good idea to test whether model without trend, too, would reject.) 21 Power of Augmented Dickey-Fuller Test  Simulations have shown that the power of (A)DF test is relatively low  The test accepts the null hypothesis of unit root too often  A critical ”eye” should be used when making conclusions  Especially in cases with relatively small number of observations  For instance, if the true DGP of time series is ADF test does not work well, if the nobs is relatively small  If the null hypothesis cannot be rejected, either the time series is truly non-stationary OR there is just not enough evidence against the null hypothesis to reject it (or there could be structural breaks in the time series) 22 On Unit Root Tests  An alternative unit root test, Phillips-Perron (PP) test, suffers from similar complications  In practice, the researcher (at least an experienced one) usually has a strong a priori assumption of the order of integration of a series, i.e., whether the series is I(0), I(1) or even I(2)  However, sometimes the test results are surprising, in which case it may be necessary to continue the investigation e.g. with alternative unit root tests  The same applies to ’bordeline cases’  In borderline cases, one can examine whether the result is robust to lag length selection (i.e. whether the conclusion is dependent on p) 23 On Unit Root Tests: Structural Breaks  A common complication in unit root tests (not only in ADF) are possible structural changes/breaks in the time series  If there is a structural change, the tests indicate a unit root too often  For instance, the (hypothetical) time series graphed below: • yt is stationary in both sides of the structural break occuring in date 50 • Nevertheless, unit root test tends to accept the null hypothesis of non-stationarity 24 Empirical Example (data included in Moodle!)  Eviews (in the variable window): Unit root tests – Standard unit root tests  Daily Neste share total return index (to allow for possible trend stationarity, we may add trend in the test for level)   Accurate p-values are provided by the MacKinnon (1996) surfaceresponse tables (here SIC for lag length, but could use MAIC instead) When the difference (i.e. return) is tested, the model is: 2yt = 1yt-1 + t (+ c + bt) Null Hypothesis: D(LN_RET) has a unit root Exogenous: Constant Lag Length: 0 (Automatic - based on SIC, maxlag=10) Null Hypothesis: LN_RET has a unit root Exogenous: Constant, Linear Trend Lag Length: 0 (Automatic - based on SIC, maxlag=10) Augmented Dickey-Fuller test statistic Test critical values: 1% level 5% level 10% level t-Statistic Prob.* -0.757296 -3.960335 -3.410929 -3.127272 0.9679 Augmented Dickey-Fuller test statistic 1% level Test critical values: 5% level 10% level t-Statistic Prob.* -63.13825 -3.431802 -2.862067 -2.567094 0.0001 *MacKinnon (1996) one-sided p-values. *MacKinnon (1996) one-sided p-values. Augmented Dickey-Fuller Test Equation Dependent Variable: D(LN_RET) Method: Least Squares Date: 09/27/21 Time: 15:42 Sample (adjusted): 4/19/2005 9/11/2020 Included observations: 3997 after adjustments (POLL) Augmented Dickey-Fuller Test Equation Dependent Variable: D(LN_RET,2) Method: Least Squares Date: 09/27/21 Time: 15:45 Sample (adjusted): 4/20/2005 9/11/2020 Included observations: 3996 after adjustments Note: Dependent variable is the 2nd difference, when the 1st difference is tested Variable Coefficient Std. Error t-Statistic Prob. Variable Coefficient Std. Error t-Statistic Prob. LN_RET(-1) C @TREND("4/18/2005") -0.000472 0.000193 7.78E-07 0.000623 0.001053 4.48E-07 -0.757296 0.183556 1.737989 0.4489 0.8544 0.0823 D(LN_RET(-1)) C -0.998925 0.000656 0.015821 0.000354 -63.13825 1.854095 0.0000 0.0638 25 Empirical Example  Monthly U.S. default risk premium Null Hypothesis: D(RISKPREMIUM) has a unit root Exogenous: Constant Lag Length: 0 (Automatic - based on Modified AIC, maxlag=12) Augmented Dickey-Fuller test statistic Test critical values: 1% level 5% level 10% level t-Statistic Prob.* -9.570428 -3.456302 -2.872857 -2.572875 0.0000 *MacKinnon (1996) one-sided p-values. Without constant: unit root not rejected With constant: borderline case, rejected at the 9% level  Which should we rely on? (POLL) 26 Alternative Unit Root Tests  DF-GLS (Dickey-Fuller Generalized Least Squares)  Elliot et al. (1996)  Same basic principle as in the ADF test, but slightly modified  Same as ADF, if no deterministic variables included  If deterministic variables included, DF-GLS is more powerful than ADF  Phillips-Perron test (PP test; Phillips & Perron, 1988)  Also a version that caters for possible structural break exists  KPSS  Kwiatkowski et al. (1992)  Null hypothesis is that of stationarity!  Several other tests also exist 27 Empirical Example: Alternative Tests  Monthly U.S. default risk premium; PP and KPSS tests KPSS rejects the null of stationarity: it is clearly ”safer” to conclude that risk premium is a non-stationary series Note: KPSS is not available without constant Both tests clearly indicate stationarity of the 1st difference 28 What to Do If a Variable Is Non-Stationary  In univariate analysis, such as ARMA models:   If trend stationary, i.e. if rejection of unit root in ADF (or PP) test requires the presence of constant, remove deterministic trend OR include deterministic trend as an explanatory (RHS) variable For difference stationary series: take the difference  Modelling with stationary series  Multivariate analysis     Testing for cointegration if more than one I(1) series If e.g. two I(1) series that are cointegrated: differencing is not needed If no cointegration: transforming the variables to be I(0) Overdifferencing should be avoided: information is always lost, when taking differences 29