Presentation on theme: "Unit Root Tests: Methods and Problems"— Presentation transcript:
1 Unit Root Tests: Methods and Problems Roger PermanApplied Econometrics Lecture 12
2 How do you find out if a series is stationary or not? Unit Root TestsHow do you find out if a series is stationary or not?
3 Y = b0 + Y + e ® I (1) D Y = Y - Y = b0 + e ® I (0) Order of Integration of a SeriesA series which is stationary after being differenced onceis said to be integrated of order 1 and is denoted by I(1).In general a series which is stationary after beingdifferenced d times is said to be integrated of order d,denoted I(d). A series, which is stationary withoutdifferencing, is said to be I(0)Y=b0+Y+eI(1)tt-1tDY=Y-Y=b0+eI(0)ttt-1t
4 Informal Procedures to identify non-stationary processes (1) Eye ball the data (a) Constant mean?(b) Constant variance?
5 Informal Procedures to identify non-stationary processes (2) Diagnostic test - CorrelogramCorrelation between 1980 and k.For stationary process correlogram dies out rapidly.Series has no memory is not related to 1985.
6 Informal Procedures to identify non-stationary processes (2) Diagnostic test - CorrelogramFor a random walk the correlogram does not die out.High autocorrelation for large values of k
7 Statistical Tests for stationarity: Simple t-test Set up AR(1) process with drift (b0)Yt = b0 + b1Yt-1 + t t ~ iid(0,σ2) (1)Simple approach is to estimate eqn (1) using OLS and examine estimated b1Use a t-test with null Ho: b1 = 1 (non-stationary)against alternative Ha: b1 < 1 (stationary).Test Statistic: TS = (b1 – 1) / (Std. Err.(b1))reject null hypothesis when test statistic is large negative- 5% critical value is -1.65
8 Statistical Tests for stationarity: Simple t-test Simple t-test based on AR(1) process with drift (b0)Yt = b0 + b1Yt-1 + t t ~ iid(0,σ2) (1)Problem with simple t-test approach(1) lagged dependent variables => b1 biased downwards in small samples (i.e. dynamic bias)(2) When b1 =1, we have non-stationary process and standard regression analysis is invalid(i.e. non-standard distribution)
9 Dickey Fuller (DF) approach to non- stationarity testing Dickey and Fuller (1979) suggest we subtract Yt-1 from both sides of eqn. (1)Yt - Yt-1 = b0 + b1Yt-1 - Yt-1 + t t ~ iid(0,σ2)ΔYt = b0 + Yt-1 + t = b1 – (2)Use a t-test with: null Ho: = 0 (non-stationary or Unit Root)against alternative Ha: < 0 (stationary).- Large negative test statistics reject non- stationarity- This is known as unit root test since in eqn. (1) Ho: b1 =1.
10 The difference between the three regressions concerns the Variants of DF testThree different regression can be used to test the presence of a unit rootThe difference between the three regressions concerns thepresence of deterministic elements b0 and b2t.1 – For testing if Y is a pure Random Walk2 – For testing if Y is a Random Walk with Drift3 – For testing if Y is a Random walk with Drift and Deterministic Trend
11 The simplest model (appropriate only if you think there are no other terms present in the ‘true’ regression model)Use the t statistic and compare it with the the table of critical values computed by Dickey and Fuller. If your t value is outside the confidence interval, the null hypothesis of unit root is rejectedStatistic
12 A more general model (allowing for ‘drift’) Statistic - Use the F statistic to check if = b0 = 0 using thenon standard tablesStatistic - use the t statistic to check if =0 , again usingnon-standard tables
13 ExampleSample size of n = 25 at 5% level of significance for eqn. (2)τμ-critical value = t-test critical value = -1.65Δpt-1 = pt-1(-1.05) (-1.49) = τμ = > -3.00hence cannot reject H0 and so unit root.
14 Incorporating time trends in DF test for unit root Some time series clearly display an upward or downward trend (non-stationary mean).Should therefore incorporate trend in the regression used for the DF test.ΔYt = b0 + Yt-1 + b2 trend + t (4)It may be the case that Yt will be stationary around a trend. Although if a trend is not included series is non-stationary.
17 Summary of Dickey-Fuller Tests (Critical values for n = 100)
18 Augmented Dickey Fuller (ADF) test for unit root Dickey Fuller tests assume that the residuals t in the DF regression are non- autocorrelated.Solution: incorporate lagged dependent variables.For quarterly data add up to four lags.ΔYt = b0 + Yt-1 + θ1ΔYt-1 + θ2ΔYt-2 + θ3ΔYt-3 + θ4ΔYt-4 + t (3)Problem arises of differentiating between models.Use a general to specific approach to eliminate insignificant variablesCheck final parsimonious model for autocorrelation.Check F-test for significant variablesUse Information Criteria. Trade-off parsimony vs. residual variance.
19 Consider The Following Series and Its Correlogram This variable Y is clearly trended and you have to determine if this trend is stochasticor deterministic. After having created the difference variable Y estimate thefollowing model, with as many lags of Y as you think appropriate.(in the example I choose 4 lags of the variable Y)
23 Choose Between Alternative Models - The Model-Progress Results Both the F-Test and the Schwarz Information Criteria indicatesthat MODEL 4 is the one to be preferred
24 To do this perform an F-Test and use the statistic Unit Root TestingAfter having estimated, according to the previous analysis, the following equationthe relevant hypotheses to examine are (in this particular case)H:b,b,b=b,,2vH:b,b,bb,,12To do this perform an F-Test and use the statistic
25 PcGive output of test result: Wald test for linear restrictions: SubsetLinRes F( 2,493) = [0.0066] **Be careful here. The value is not significant at the 5% critical value, although PcGive marks it as significant (it is using the conventional F distribution).Therefore we cannot reject the null hypothesis, and so infer that we do not have a deterministic time trend in the equation. Hence, we can continue the analysis usingand then usestatistic - use the F statistic to check if = b0 = 0 using thenon standard tablesstatistic - use the t statistic to check if =0 , again usingnon-standard tables
26 The t-stat cannot reject the null hypothesis of Unit Root while the F-stat rejects the null hypothesis that the drift is equal to zero. Therefore we canconclude that the model most likely to describe the true DGP is
27 Reject Accept Reject Accept Reject Accept Reject Accept Look at the Series – Is there a Trend?YesNoEstimateEstimateUseUseto testto testH:b,b=,H:b,b,b=b,,2vvH:b,b,b1H:b,b,b,,12RejectAcceptRejectAccepttest =0 using thet-stat. from step 1usingPure Random Walktest =0 using thet-stat. from step 1usingRejectAcceptRejectAcceptNo Unit RootUnit Root +TrendStable Series,use normal testto check the driftRandomWalk + DriftUseNormal Test procedureto determine thepresence ofTime trend or DriftTo determine if thereis a drift as well
28 Alternative statistical test for stationarity One further approach is the Sargan and Bhargava (1983) test which uses the Durbin-Watson statistic.If Yt is regressed on a constant alone, we then examine the residuals for serial correlation.Serial correlation in the residuals (long memory) will fail the DW test and result in a low value for this test.This test has not proven so popular.
29 Testing Strategy for Unit Roots Three main aspects of Unit root testing- Deterministic components (constant, time trend).- ADF Augmented Dickey Fuller test - lag length- use F-test or Schwarz Information Criteria- In what sequence should we test?- Phi and tau tests
30 Testing Strategy for Unit Roots Formal Strategy(A) Set up Model(1) Use informal tests – eye ball data and correlogram(2) Incorporate Time trend if data is upwards trending(3) Specification of ADF test– how many lags should we incorporate to avoid serial correlation?
31 Example- Real GDP (2000 Prices) Seasonally Adjusted (1) Plot Time Series - Non-Stationary(i.e. time varying mean and correlogram non-zero)GDPTimerk
32 (1) Plot First Difference of Time Series - Stationary Unit Root Testing(1) Plot First Difference of Time Series - Stationary(i.e. constant mean and correlogram zero)Timerk
33 Unit Root Testing(2) Incorporate Linear Trend since data is trending upwards
34 (3) Determine Lag length of ADF test Unit Root Testing(3) Determine Lag length of ADF testEstimate general model and test for serial correlationEQ ( 1) ΔYt = b0 +b2 trend+ Yt-1 + θ1ΔYt-1 + θ2ΔYt-2 + θ3ΔYt-3 + θ4ΔYt-4 + tEQ( 1) Modelling DY by OLS (using Lab2.in7)The estimation sample is: 1956 (2) to 2003 (3) n = 190Coefficient Std.Error t-value t-prob Part.R^2ConstantTrendY_DY_DY_DY_DY_AR 1-5 test: F(5,178) = [0.1308]Test accepts null of no serial correlation.Nevertheless we use F-test and Schwarz Criteria to check model.
35 (3) Determine Lag length of ADF test Unit Root Testing(3) Determine Lag length of ADF testModelEQ ( 1) ΔYt = b0+b2 trend+ Yt-1 + θ1ΔYt-1 + θ2ΔYt-2 + θ3ΔYt-3 + θ4ΔYt-4 + tEQ ( 2) ΔYt = b0+b2 trend+ Yt-1 + θ1ΔYt-1 + θ2ΔYt-2 + θ3ΔYt-3 + tEQ ( 3) ΔYt = b0+b2 trend+ Yt-1 + θ1ΔYt-1 + θ2ΔYt-2 + tEQ ( 4) ΔYt = b0+b2 trend+ Yt-1 + θ1ΔYt-1 + tEQ ( 5) ΔYt = b0+b2 trend+ Yt-1 + tUse both the F-test and the Schwarz information Criteria (SC).Reduce number of lags where F-test accepts.Choose equation where SC is the lowesti.e. minimise residual variance and number of estimated parameters.
36 (3) Determine Lag length of ADF test Unit Root Testing(3) Determine Lag length of ADF testProgress to dateModel T p log-likelihood Schwarz CriteriaEQ( 1) OLSEQ( 2) OLSEQ( 3) OLSEQ( 4) OLSEQ( 5) OLSTests of model reductionEQ( 1) --> EQ( 2): F(1,183) = [0.5259] Accept model reductionEQ( 1) --> EQ( 3): F(2,183) = [0.0357]* Reject model reductionEQ( 1) --> EQ( 4): F(3,183) = [0.0173]*EQ( 1) --> EQ( 5): F(4,183) = [0.0374]*Some conflict in results. F-tests suggest equation (2) is preferred to equation (1) and equation (3) is not preferred to equation (2).Additionally, the relative performance of these three equations is confirmed by information criteria.Therefore adopt equation (2).
37 (B) Conduct Formal Tests Unit Root Testing(B) Conduct Formal TestsEQ( 2) Modelling DY by OLS (using Lab2.in7)The estimation sample is: 1956 (2) to 2003 (3)Coefficient Std.Error t-value t-prob Part.R^2ConstantTrendY_DY_DY_DY_AR 1-5 test: F(5,179) = [0.6357]Main issue is serial correlation assumption for this test.Can we accept the null hypothesis of no serial correlation? Yes!
38 Unit Root TestingApply F-type test - Include time trend in specificationΦ3: ΔYt = b0 + b2 trend + Yt-1 + θ1ΔYt-1 + θ2ΔYt-2 + θ3ΔYt-3 + t (a) Ho: = b2 = 0 Ha: β 0 and/or b2 0PcGive Output : Test/Exclusion Restrictions.Test for excluding:  = Trend  = Y_1F(2,184) = < 6.39 = 5% C.V. (by interpolation).Hence accept joint null hypothesis of unit root and no time trend(next test whether drift term is required).NB Critical Values (C.V.) from Dickey and Fuller (1981) for Φ3Sample Size (n)C.V. at 5%
39 Φ1: ΔYt = b0 + Yt-1 + θ1ΔYt-1 + θ2ΔYt-2 + θ3ΔYt-3 + t Unit Root TestingApply F-type test - Exclude time trend from specificationΦ1: ΔYt = b0 + Yt-1 + θ1ΔYt-1 + θ2ΔYt-2 + θ3ΔYt-3 + t(b) Ho: = b0 = 0 Ha: 0 and/or b0 0PcGive Output : Test/Exclusion Restrictions.Test for excluding:  = Constant = Y_1F(2,185) = > 4.65 = 5% C.V.Hence reject joint null hypothesis of unit root and no drift.NB Critical Values (C.V.) from Dickey and Fuller (1981) for Φ1Sample Size (n)C.V. at 5%
40 τμ ΔYt = b0 + Yt-1 + θ1ΔYt-1 + θ2ΔYt-2 + θ3ΔYt-3 + t Unit Root TestingApply t-type test (τμ)τμ ΔYt = b0 + Yt-1 + θ1ΔYt-1 + θ2ΔYt-2 + θ3ΔYt-3 + t(b) Ho: = 0 Ha: < 0τμ = > = 5% C.V.Hence accept null of unit root.N.B. Critical Values (C.V.) from Fuller (1976) for τμSample Size (n)C.V. at 5%
41 Unit Root TestingEQ(2a) Modelling DY by OLS (using Lab2.in7)The estimation sample is: 1956 (2) to 2003 (3)Coefficient Std.Error t-value t-prob Part.R^2ConstantY_DY_DY_DY_AR 1-5 test: F(5,180) = [0.7725]τμ = > (5% C.V.) hence we can not reject the null of unit root.
43 Problem Number 1: Structural Breaks Perron (1989) - He argues that most macroeconomic variablesare not unit root processes. They are Trend Stationary withStructural BreaksFor example1929 DepressionOil ShocksTechnological ChangeAll these events have changed the mean of a process like GDPIf you do not recognize the structural break, you’ll find unit rootwhere there is notWith Structural Change All Unit Root Tests Are BiasedTowards the Non Rejection of a Unit Root
44 The Unit Root Hypothesis is not rejected. Unit Root Test For YDickey-Fuller test for Y; DY onVariable Coefficient Std.Error t-valueY_\sigma = DW = DW(Y) = DF(Y) =Critical values used in DF test: 5%= %=-2.587RSS = for 1 variables and 98 observationsInformation Criteria:SC = HQ = FPE= AIC =The Unit Root Hypothesis is not rejected.Perron proposed a method to overcome this problem - But youneed to know when the structural break happened
45 Consider The Following Variable Y=.5Y+D+ett-1tfort=..49D=t2fort=50..100
46 The Power of a test is the probability of rejecting a false Problem Number Two : Low PowerThe Power of a test is the probability of rejecting a falseNull Hypothesis -Unit Root TestsLow Power to Distinguish Between Unit and near Unit RootLow Power to Distinguish Between Trend and DriftY is a unit root series;Z is a near-unit root series
47 Test result is based on the standard error of Is = 0 in ΔYt = b0 + Yt-1 + tTest result is based on the standard error of - Measure of how accurate is our estimated coefficient- with increasing observations we become more certain.In this case, power of the test is the ability to reject the null of non-stationarity when it is false (equivalently, the ability to accept alternative hypothesis of stationarity).Low power implies a series may be stationary but Dickey-Fuller test suggests unit root.Low power is especially a problem when series is stationary but close to being unit root.One solution to low power is to increase the number of observations by increasing the span of data. However, there may be differences in economic structure or policy which should be modelled differently. Alternative solution to low power is a number of joint ADF tests.- Take information from a number of countries.- And pool coefficients. (i.e. combine information).