7. Integrated Processes

Transcription

1 7. Integrated Processes Up to now: Analysis of stationary processes (stationary ARMA(p, q) processes) Problem: Many economic time series exhibit non-stationary patterns over time 226

2 Example: We consider the log DAX values from Slide 15 We estimate the AR(1) process DAX t = c + φ DAX t 1 + ɛ t Obviously: ˆφ = close to 1 (Slide 228) DAX process is likely to have a unit root (AR polynomial φ(l) = 1 L has a root on the unit circle; unit root) DAX process is a random walk and non-stationary 227

3 Estimation of an AR(1) process for the log DAX Dependent Variable: DAX_LOG Method: Least Squares Date: 22/06/08 Time: 13:39 Sample (adjusted): 1960M M02 Included observations: 578 after adjustments Variable Coefficient Std. Error t-statistic Prob. C DAX_LOG(-1) R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic)

4 Remark: We obtain a similar result when fitting an ARMA(p, q) specification to the data Phrasing: When we say A time series has a unit root, we mean: The time-series process is non-stationary, but stationarity can be obtained by taking (first) differences. 229

5 .2 DAX (1st differences in log values) Time Dependent Variable: DAX_LOG_DIFF Method: Least Squares Date: 22/06/08 Time: 14:33 Sample (adjusted): 1960M M02 Included observations: 577 after adjustments Variable Coefficient Std. Error t-statistic Prob. C DAX_LOG_DIFF(-1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

6 Taking differences: 1st differences: X t = (1 L)X t = X t X t 1 2nd differences: 2 X t = (1 L) 2 X t = (1 2L + L 2 )X t = X t 2X t 1 + X t 2 Differences of order d: d X t = (1 L) d X t 231

7 Definition 7.1: (Integration of a process) We say that the process {X t } is integrated of order d [in symbols: X t I(d)], if it needs differentiation of order d to render {X t } stationary. Remarks: Most economic unit-root processes become stationary after taking first differences (i.e. the processes are I(1)) We denote stationary processes by I(0) 232

8 7.1 Stochastic Versus Deterministic Trends Remark: The non-stationarity of an ARMA(p, q) process {X t } due to a unit root refers to the roots of the AR(p) polynomial we restricit attention to the AR(p) component of {X t } (the MA(q) component is always stationary) 233

9 Summary: Many economic time series exhibit a trend (Slides vgl. Folien 9-16) AR(p) models with a unit root may capture a trend Question: Besides unit-root AR(p) processs, are there other theoretical models implying a time-trend behavior? 234

10 Consider the following model: where {ɛ t } WN(0, σ 2 ) Special cases: (I) X t = c + φ X t 1 + δ t + ɛ t, Random walk with/without drift, i.e. φ = 1, δ = 0: X t = c + X t 1 + ɛ t {X t } has a unit root (implying a trend), but first differences yield stationarity: stochastic trend X t = X t X t 1 = c + ɛ t 235

11 Special cases: (II) Trend-stationary process, i.e. φ = 0, δ 0: X t = c + δ t + ɛ t We have: E(X t ) = c + δ t, Var(X t ) = Var(ɛ t ) = σ 2 i.e. {X t } is non-stationary and has a trend. However, the process {Y t } = {X t E(X t )} = {X t c δ t} = {ɛ t } is stationary deterministic trend 236

12 Stochastic vs. deterministic trend 40 stochastic trend deterministic trend t 237

13 Features of both trends: (I) Deterministic trend: if the parameters c and δ are known, we can perfectly forecast the mean of X t deviations from the trending linie c+δ t are purely random and do not have any impact on the long-term dynamics of X t Detrending (subtracting the mean c+δ t from X t ) renders the process stationary 238

14 Features of both trends: (II) Stochastic trend: stochastic component ɛ t has a long-term impact on X t (with rigorous explanation) stationarity is achieved via taking differences 239

15 7.2 Hypothesis testing in AR(p) Models With Deterministic Trend General setting: We consider the AR(p) model X t = c + φ 1 X t φ p X t p + ɛ t with ɛ t WN(0, σ 2 ) Objective: Construction of a statistical test for detecting a unit root in a time series (test for stationarity) 240

16 Preliminary step: Convenient reformulation of the AR(p) model Theorem 7.2: (Equivalent representation of an AR(p) process) Let {X t } be an AR(p) process according to the original definition repeated above. Then, using the appropriately defined parameters ϱ, ψ 1,..., ψ p 1, we can equivalently represent {X t } as follows: X t = c+ϱ X t 1 +ψ 1 X t 1 +ψ 2 X t ψ p 1 X t p+1 +ɛ t. 241

17 Remarks: The equivalence of both AR(p) representations follows from some straight forward algebraic manipulations The new parameters ϱ, ψ 1,..., ψ p 1 are simple functions of the original parameters φ 1,..., φ p For example: ϱ = φ 1 + φ φ p 1 Each representation contains p + 1 parameters, namely c, φ 1,..., φ p and c, ϱ, ψ 1,..., ψ p 1 242

18 Theorem 7.3: (Stationarity of AR(p) processes) Let {X t } be an AR(p) process and consider its representation from Theorem 7.2. We then have: (a) The process {X t } is stationary, if 2 < ϱ < 0. (b) The process {X t } has a unit root (i.e. it is non-stationary), if ϱ =

19 Remarks: In the AR(p) representation from Theorem 7.2 the stationarity solely depends on the parameter ϱ In the original representation stationarity hinges on the absolute values of the roots of the AR(p) polynomial Φ(z) = 1 φ 1 z φ 2 z 2... φ p z p and thus on the p parameters φ 1,..., φ p (cf. Slides 84, 85) 244

20 Now: Consider the AR(p) model from Theorem 7.2 extended to cover a deterministic trend: X t = c + ϱ X t 1 + p 1 j=1 ψ j X t j + δ t + ɛ t (AR(p) model with deterministic trend, for short: AR(p)-DT model) Test for a unit root in the AR(p)-DT model: H 0 : ϱ = 0 versus H 1 : ϱ < 0 245

21 Our concern: Statistical tests in the AR(p)-DT model Two objectives: Selection of the appropriate lag length p Test for a unit root To this end: Consider the two groups of parameters (a) c, ψ 1,..., ψ p 1 and δ (b) ϱ 246

22 Justification: Test for the parameter ϱ is different from the tests for all other parameters (exact explanation will follow) Now: Sequential procedure for selecting the lag length in the AR(p)- DT model 247

23 Five-step-strategy: (I) 1. Choose a reasonable maximal lag length p max 2. Estimate the AR(p max )-DT model by OLS X t = c + ϱ X t 1 + p max 1 j=1 Consider the conventional t test: ψ j X t j + δ t + ɛ t H 0 : ψ pmax 1 = 0 versus H 1 : ψ pmax 1 0 In case of the rejection of H 0, proceed with Step #5 In case of the non-rejection of H 0, proceed with Step #3 248

24 Five-step-strategy: (II) 3. Estimate the AR(p max 1)-DT model by OLS X t = c + ϱ X t 1 + p max 2 j=1 Consider the conventional t test: ψ j X t j + δ t + ɛ t H 0 : ψ pmax 2 = 0 versus H 1 : ψ pmax 2 0 In case of the rejection of H 0, proceed with Step #5 In case of the non-rejection of H 0, proceed with Step #4 4. Repeat estimation of the model performing stepwise lag reduction until either a ψ-coefficient gets statistically significant or until there is no more lag left to reduce 249

25 Five-step-strategy: (III) 5. Use the t test H 0 : δ = 0 versus H 1 : δ = 0 and decide on whether to retain the deterministic trend in the model 250

26 Example: We consider the log. DAX time series and start with p max = 4 Result (see Slides 252, 253): Model reduces to X t = c + ϱ X t 1 + δ t + ɛ t Question: Is ϱ = 0, that is, does the time series have a unit root? Plausible procedure: The conventional t test indicates that ϱ is significantly different from zero at the 5% level (see Slide 253) 251

27 Dependent Variable: D(DAX_LOG) Method: Least Squares Date: 22/06/08 Time: 17:42 Sample (adjusted): 1960M M02 Included observations: 575 after adjustments Variable Coefficient Std. Error t-statistic Prob. C DAX_LOG(-1) D(DAX_LOG(-1)) D(DAX_LOG(-2)) D(DAX_LOG(-3)) T 8.96E E R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Dependent Variable: D(DAX_LOG) Method: Least Squares Date: 22/06/08 Time: 17:43 Sample (adjusted): 1960M M02 Included observations: 576 after adjustments Variable Coefficient Std. Error t-statistic Prob. C DAX_LOG(-1) D(DAX_LOG(-1)) D(DAX_LOG(-2)) T 8.79E E R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

28 Dependent Variable: D(DAX_LOG) Method: Least Squares Date: 22/06/08 Time: 17:45 Sample (adjusted): 1960M M02 Included observations: 577 after adjustments Variable Coefficient Std. Error t-statistic Prob. C DAX_LOG(-1) D(DAX_LOG(-1)) T 8.71E E R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Dependent Variable: D(DAX_LOG) Method: Least Squares Date: 22/06/08 Time: 17:46 Sample (adjusted): 1960M M02 Included observations: 578 after adjustments Variable Coefficient Std. Error t-statistic Prob. C DAX_LOG(-1) T 8.45E E R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

29 Unfortunately: In the AR(p)-DT model the conventional t test on the parameter ϱ is invalid In the AR(p)-DT model the null distribution of the t statistic of the parameter ϱ does not follow the t distribution, but follows an entirely different distribution 254

30 7.3 Statistical Tests for Unit Roots Question: How can we test for a unit root (i.e. for non-stationarity) of a time series? To this end: We consider the AR(p)-DT model X t = c + ϱ X t 1 + p 1 j=1 ψ j X t j + δ t + ɛ t, that we have estimated following the five-step-strategy from Slides

31 Recall: The statistical test is equivalent to H 0 : ϱ = 0 versus H 1 : ϱ < 0 H 0 : time series has a unit root (stochastic trend) versus H 1 : time series does not have a unit root (no stochastic trend) 256

32 Recall: In the AR(p)-DT model, the conventional t statistic of the parameter ϱ does not follow a t distribution under H 0 (Slide 254) Therefore: In the AR(p)-DT model the t statistic of the parameter ϱ is called τ statistic 257

33 Remarks on the null distribution of the τ statistic: The distribution is due to Dickey-Fuller (1979, 1981) The exact form of the null distribution of the τ statistic crucially hinges on the inclusion of the deterministic trend t in the AR(p)-DT model the inclusion of the constant c in the AR(p)-DT model The critical values of the null distribution are called MacKinnon values and are implemented in all econometric software packages 258

34 Definition 7.4: (Augmented-Dickey-Fuller test) We consider the AR(p)-DT model from Slide 245, which we fit to time-series data according to the five-step-strategy described on Slides The hypothesis test concerning the problem H 0 : ϱ = 0 versus H 1 : ϱ < 0 based on the τ statistic and the MacKinnon critical values is called Augmented-Dickey-Fuller (ADF) test. 259

35 Remarks: The ADF test is a test for stationarity The ADF test rejects the null hypothesis of a unit root (i.e. the non-stationarity) in favor of the alternative (the stationarity), if the computed τ value is more negative than the critical MacKinnon value at the considered significance level (i.e. if the absolute τ value is larger than the absolute Mac- Kinnon value) 260

36 Example: (ADF test for the log. DAX) The five-step-strategy from Slides yields an AR(p)- DT model with p = 1, δ = 0 (model without lagged differences, with deterministic trend) Null Hypothesis: DAX_LOG has a unit root Exogenous: Constant, Linear Trend Lag Length: 0 (Automatic based on SIC, MAXLAG=18) t-statistic Prob.* Augmented Dickey-Fuller test statistic Test critical values: 1% level % level % level *MacKinnon (1996) one-sided p-values. 261

37 Results in EViews: Computed τ value = in absolute terms is smaller than the respective absolute values of the 3 critical MacKinnon values H 0 cannot be rejected at the 1%, 5%, 10% levels indication of a unit root (stochastic trend) 262

38 Final remarks: (I) The ADF test assumes the following properties of the error process {ɛ t } t=0,1,... in the AR(p)-DT model as satisfied: the error terms ɛ t are stochastically independent (absence of autocorrelation) the error terms are homoscedastic (Var(ɛ t ) = σ 2 for all t) An alternative (non-parametric) unit-root test, that accounts for autocorrelation and heteroscedasticity in the error process {ɛ t }, is the Phillips-Perron test 263

39 Final remarks: (II) In practice, ADF tests often have low power, i.e. ADF tests too often retain the null hypothesis of a unit root (hence the non-stationarity), if in fact the null hypothesis is false (error-type II of the test) If the time-series data are subject to a structural break, then the ADF test results are completely unreliable 264

40 7.4 Regressions With Integrated Variables Up to now: Properties of the stochastic process {X t }: stationarity unit root (integrated process) Now: Regression equations with time-series variables, e.g. Y t = β 0 + β 1 X 1t β K X Kt + ɛ t 265

41 Important issue: Properties of the OLS estimators of β 0,..., β K, if all variables are stationary some variables contain a unit root problem of spurious regressions Important result: If all time-series variables {Y t }, {X 1t },..., {X Kt } are stationary and the classical assumptions of the linear regression model are satisfied, then OLS estimation is unproblematic 266

42 7.4.1 Spurious Regressions Question: Why are regressions containing unit-root variables problematic? 267

43 Example: Consider two random-walk variables {Y t }, {X t } (non-stationary processes containing a unit root) Both processes are independently generated: Y t = Y t 1 + ɛ 1t X t = X t 1 + ɛ 2t with {ɛ 1t }, {ɛ 2t } GWN(0, 1) ({ɛ 1t }, {ɛ 2t } independent) Consider the regression specification in levels in 1st differences Y t = β 0 + β 1 X t + ɛ t Y t = β 0 + β 1 X t + ɛ t 268

44 60 1st Random Walk nd Random Walk Dependent Variable: RANDOMWALK_1 Method: Least Squares Date: 01/20/03 Time: 10:10 Sample: Included observations: 1000 t Variable Coefficient Std. Error t-statistic Prob. C RANDOMWALK_ R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

45 4 1st differences of 2nd random walk st differences of 1st random walk Dependent Variable: D(RANDOMWALK_1) Method: Least Squares Date: 01/20/03 Time: 10:14 Sample(adjusted): Included observations: 999 after adjusting endpoints Variable Coefficient Std. Error t-statistic Prob. C D(RANDOMWALK_2) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

46 Regression in levels: Parameter β 1 is statistically significant R 2 is high and also statistically significant (see F test) Regression in 1st differences: Parameter β 1 is no longer significant R 2 is virtually zero and also no longer significant 271

47 Summary: For the regression in levels, the (significant) relationship solely results from the stochastic trends inherent in both time series (spurious regression) The regression in 1st differences indicates that both variables are uncorrelated (consistency with the data-generating process) 272

48 7.4.2 Cointegration Now: We simplify and consider the two-variable regression Y t = β 0 + β 1 X t + ɛ t with {Y t } and {X t } having both a unit root Question: Are there situations, in which the problem of spurious regressions does not emerge? 273

49 Answer: Yes, if {Y t } and {X t } are cointegrated despite the unit roots, OLS estimation is unproblematic Definition 7.5: (Cointegration) Let {X t }, {Y t } I(1) be two unit-root processes. If there is a coefficient β such that the process {Y t βx t } is stationary (i.e. if {Y t βx t } I(0)), then {X t } and {Y t } are said to be cointegrated. We call β the cointegration coefficient. 274

50 Remarks: If {X t } and {Y t } are cointegrated, then both processes have a joint stochastic trend the difference process {Y t βx t } eliminates the joint stochastic trend A cointegrating relationship between {Y t } and {X t } represents a long-term equilibrium link among the processes 275

51 Economic examples of cointegration: (I) 1. Prices for substitutable goods (e.g. prices for regular and bio oranges) Reason for cointegration: In principle, consumers have higher willingness-to-pay for bio oranges (price curve of bio oranges is higher) However, the higher willing-to-pay is limited distance between both price curves should be stationary 276

52 Prices per pound of regular and bio oranges 300 Price per pound Bio oranges Regular oranges Months 277

53 Economic examples of cointegration: (II) 2. Short- and long-term interest rates Long-term interest rates are often higher than short-term rates (risk premia, term structure of interest rates) However, too high risk premia induce investors to restructure their portfolios interest rates approach again due to market mechanism 278

54 Economic examples of cointegration: (III) 3. (Absolute) purchasing power parity Nominal exchange rate reflects purchasing power between two countries, i.e. W t = P t, or in logs, w t = p t p t with P t W = nominal exchange rate P, P = domestic and foreign price levels relationship holds in equilibrium (with adjustment reactions, if not satisfied) {w t + p t p t} should be stationary (i.e. the processes {w t }, {p t p t } are cointegrated) 279

55 Theorem 7.6: (Consequences of cointegration) We consider the two-variables regression Y t = β 0 + β 1 X t + ɛ t. If {X t } and {Y t } are cointegrated, the following holds: 1. The problem of spurious regression does not occur. 2. OLS estimation of the two-variables regression yields valid statistical inference. 280

56 7.4.3 A Test for Cointegration Aim: Establishing a statistical test for cointegration between two I(1) processes {X t }, {Y t } Intuition: We may base the test on the two-variables regression: Y t = β 0 + β 1 X t + ɛ t Y t β 1 X t }{{} difference process = β 0 + ɛ t Engle-Granger cointegration test in three steps 281

57 Engle-Granger cointegration test: 1. We estimate the two-variables regression by OLS and save the residuals 2. We apply the ADF test for stationarity (omitting the deterministic trend) to the residuals (see Section 7.3) 3. If the ADF test rejects the null hypothesis of a unit root, we infer cointegration between {X t } and {Y t }. If the ADF test does not reject the null hypothesis of a unit root, we cannot infer cointegration between {X t } and {Y t }. 282

58 Remarks: (I) To conduct the ADF test for stationarity, we have to fit the AR(p) model to the residuals via the five-step-strategy from Slides However, the Engle-Granger cointegration test does without the deterministic trends t Since the residuals {ˆɛ t } based on the the parameter estimates ˆβ 0 and ˆβ 1 (and not on the true unknown parameters β 0, β 1 ), we need to apply corrected critical MacKinnon values 283

59 Remarks: (II) For given sampling size T the critical values may be approximated as follows: 1% critical value = T T 2 5% critical value = T T 2 10% critical value = T T 2 (cf. MacKinnon, 1991, pp ) The Engle-Granger cointegration test rejects the null hypothesis of a unit root in the residuals (no cointegrating relationship) if the computed value of the τ statistic is more negative than the chosen critical value 284

60 Remarks: (III) The Engle-Granger test essentially is an ADF test for a unit root in the residuals the remarks on the ADF test given on Slides 263, 264 also apply here (assumptions on the error term, low power) 285

61 Example: (Link between consumption income) Consider the US time series private consumption expenditures (PCE) private disposable income (PDI) Are the processes {PCE t }, {PDI t } cointegrated? (see class) 286

62 7.4.4 Vector Error Correction Models General setting: {Y t }, {X t } I(1) are cointegrated with cointegration coefficient β This implies: { Y t } and { X t } are stationary {Y t βx t } is stationary 287

63 Consequence: We may estimate the differenced processes { Y t } and { X t } via the following two-equation model: Y t = β 10 + β 11 Y t β 1p Y t p + γ 11 X t γ 1p X t p + α 1 (Y t 1 βx t 1 ) + ɛ 1t (1) X t = β 20 + β 21 Y t β 2p Y t p + γ 21 X t γ 2p X t p + α 2 (Y t 1 βx t 1 ) + ɛ 2t (2) Definition 7.7: (Error correction model) We call the two-equation system stated above a vector error correction model and the term {Y t 1 βx t 1 } the error correction term. 288

64 Remarks: Since all variables of the two-equation system are stationary, we may estimate each equation separately by OLS (cf. Stock and Watson, 2011, p. 693) In practice, the true cointegrating coefficient β is unknown we replace the error correction term by the regressor Ẑ t 1 = Y t 1 ˆβX t 1, where ˆβ is the OLS estimator of β Past values of the error correction term {Y t βx t } are used to forecast future values of { Y t } and { X t } 289

65 Example: Interest-rate data (cf. Stock & Watson, 2011, pp ) 290

66 7.4.5 Multiple Cointegration Now: Extension of the concept to multiple regressors Y t = β 0 + β 1 X 1t β K X Kt + ɛ t Analogous approach: Under the assumption that {Y t }, {X 1t },... {X Kt } I(1), the variables are said to be cointegrated with cointegrating coefficients θ 1,... θ K, if {Y t θ 1 X 1t... θ K X Kt } I(0) 291

67 Note: In the case of multiple regressors we may have different cointegrating relationships, e.g. between {Y t } and {X 1t }, between {Y t } and {X 2t }, and so on Test for cointegration: Extended Engle-Granger test Test for multiple cointegrating relationships (cf. Johansen, 1988) 292