🗊Презентация Modeling non-stationary variables

Outline Stationary and non-stationary variables Testing for unit roots Cointegration Testing for cointegration

Introduction Many economic (macro/financial) variables exhibit trending behavior e.g., real GDP, real consumption, assets prices, dividends… Key issue for estimation/forecasting: the nature of this trend…. … is it deterministic (e.g., linear trend) or stochastic (e.g., random walk) The nature of the trend has important implications for the model’s parameters and their distributions… … and thus for the statistical procedures used to conduct inference and forecasting

Key Macro Series Appear to have trends

Deterministic and Stochastic Trends in Data Two types of trends: deterministic or stochastic A Deterministic trend is a non-random function of time Example: linear time-trend A stochastic trend is random, i.e. varies over time Examples: (Pure) Random Walk Model: a time series is said to follow a pure random walk if the change is i.i.d. Random Walk with a Drift  is a ‘drift’. If  > 0, then yt increases on average

Example: Processes with Trends

Stationary and non-stationary processes (1) Consider the data generation process (DGP) If the variable is stationary (i.e., , has finite mean and variance) Standard econometric procedures may be used to estimate/forecast this model

If model is said to be non-stationary and its associated (statistical) distribution theory is non-standard. If model is said to be non-stationary and its associated (statistical) distribution theory is non-standard. In particular: Sample moments do not have finite limits, but converge (weakly) to random quantities; Least squares estimate of is super consistent with convergence rates greater than (stationary case); Asymptotic distribution of the least squares estimator is non-standard (i.e., non-normal). Bottom line: nature of the trend has important implications for hypothesis testing and forecasting, especially in multivariate settings (e.g., VARS).

Reminder: Autoregressive AR(p) Process

Stochastic trends, autoregressive models and a unit root The condition for stationarity in an AR(p) model: roots z of the characteristic equation 1- θ1z - θ2z2 - θ3z3 - ... - θpzp =0 must all be greater than one in absolute value: |z| >1 If an AR(p) process has z=1 => variable has a unit root Example: AR(1) process yt =  + θyt-1 + vt A special case is θ =1 => z =1 => yt has unit root (stochastic trend) Stationarity requires that |θ| <1 for |z|>1

The Impact of Shocks on Stationary and Non-stationary variables Consider a simple AR(1): yt = θyt-1 + νt, where θ takes any value for now We can write: yt-1= θyt-2 + νt-1 yt-2= θyt-3 + νt-2 Substituting yields: yt = θ(θyt-2 + νt-1) + εt = θ2yt-2 + θνt-1 + νt Successive substituting for yt-2, yt-3,... gives an representation in terms of initial value y-1 and past errors νt-1, νt-2,...,ν0 yt = θt+1y-1 + θνt-1 + θ2νt-2 + θ3νt-3 + ...+ θtν0 + νt

The Impact of Shocks for Stationary and Non-stationary Series (2) Representation at t=T: yT = θT+1y-1 +θvT-1 +θ2vT-2 + θ3vT-3 + ...+ θTv0 + vT At t =0 the variable is hit by a non-zero shock v0 We have 3 cases (depending on value of θ): |θ|< 1  θT  0 and θTv0  0 as T  Shocks have only a transitory effect (gradually dies away with time) θ = 1  θT = 1 and θTv0 = v0  T Shocks have a permanent effect in the system and never die away: ... just a sum of past shocks plus some starting value of y-1. The variance grows without bound (Tσ2 ) as T |θ|>1. Now shocks become more influential as time goes on (explosive effect), since if θ>1, then |θ|T>...>|θ|3 > |θ|2 > |θ| etc.

Integration Another way to write the stochastic trend model is: Thus the first difference of yt is stationary provided vt is stationary (“difference stationary” process). Also referred to as an I(1) variable. Similarly, in the case of the deterministic trend model, yt is interpreted as trend stationary because removal of the deterministic trend from yt renders it a stationary random variable

Order of Integration: I(d) In general, if yt is I(d) then: If d=0, then the series is already stationary

Problems due to Stochastic Trends (from a statistical perspective) Non-standard distribution of test statistics Spurious regression: in a simple linear regression, two (or more) non-stationary time series may appear to be related even though they are not Need to use special modeling techniques when dealing with non-stationary data (VARs in differences or VECMs) Need to distinguish btw. stochastic and deterministic trends as it may affect estimates of policy-relevant variables e.g. estimate of an output gap or of a structural budget deficit … for that we need unit root tests…

Figure 5: Distribution of OLS estimator for θ

Testing For Unit Roots Previous section suggests that I(1) variables need special handling So how do we identify I(1) processes, i.e., test for unit roots? Natural test is to consider the t-statistic for the null-hypothesis of a unit root, i.e., Given the previous graph, it is not surprising that the t-distribution for is non-normal

Testing for Unit Roots: Procedures Dickey Fuller Augmented Dickey Fuller Phillips Perron Kwiatkowski, Phillips, Schmidt and Shin (KPSS)

Dickey Fuller Test Fuller (1976), Dickey and Fuller (1979) Example: consider a particular case of an AR(1) model: yt = θyt-1 + εt We test a hypothesis H0: θ =1 → the series contains a unit root/stochastic trend (is a random walk) against H1: |θ| <1 → the series is a zero-mean stationary AR(1)

Dickey-Fuller Test (2) For the purpose of testing we reformulate the regression: yt = yt – yt-1 =θyt-1 -yt-1 + vt = (θ-1)yt-1 + vt = = yt-1 + vt so that the test of H0: θ = 1  H0:  = 0 The test is based on the t-ratio for  this t-ratio does not have the usual t-distribution under the H0 critical values are derived from Monte Carlo experiments, and are tabulated (known): see appendix A The test is not invariant to the addition of deterministic components (more general formulation: intercept + time-trend)

Dickey-Fuller Test (3) Important issue – shall deterministic components be included in the test model for yt. Is this yt =yt-1 + vt or yt = 1+ yt-1 + vt or yt = 1+ 2t+ yt-1 + vt ? Two ways around: Use prior information/assume whether the deterministic components are included, i.e. use the restrictions (easy to implement in Eviews): 1≠0 and 2≠0 1≠0 and 2=0 1=0 and 2=0 Allow for uncertainty about deterministic components (more complicated in Eviews) and implement a testing strategy to find out: restrictions on deterministic components if yt is non-stationary

DF-Test (3): Deterministic Components are Known Say, we assume yt includes an intercept, but not a time trend yt = 1+ θyt-1 + vt We test a hypothesis: H0: θ =1 → the series has a unit root/stochastic trend against H1: |θ| <1 → the series is zero-mean stationary AR(1) Reformulate: yt = 1+ yt-1 + vt Test H0:  =0 → the series has a unit root (stochastic trend) against H1:  < 0 → the series has no unit root (is stationary) This way is easy – it is ready for you in Eviews But, there are risks involved...

DF-Test (4): Risks Posed by Deterministic Components If deterministic components are not included in the test, when they should be, then the test is not correctly sized: The test will reject the H0:  =0, although it is in fact true and should not be rejected (yt is non-stationary) – type I error If deterministic components are included but they should not be, then the test has low power (especially in finite (short) samples): The test will not reject the H0:  =0, although it is false and must be rejected (yt is stationary) – type II error This is why we may prefer (a degree of) uncertainty about deterministic components and use testing strategies (see appendix A for details): Enders Strategy Elder and Kennedy Strategy

The Augmented Dickey Fuller (ADF) Test The DF-test above is only valid if εt is a white noise: εt will be autocorrelated if there was autocorrelation in the first difference (yt), and we have to control for it The solution is to “augment” the test using p lags of the dependent variable. The alternative model (including the constant and the time trend) is now written as:

The ADF-Test (2) Again, we have three choices: (1) include neither a constant nor a time trend (2) include a constant (3) include a constant and a time trend Again, we either: use prior information and impose a model from the beginning, or remain uncertain about deterministic components and follow one of the Strategies Useful result: Critical values for the ADF-test are the same as for DF-test Note, however, that the test statistics are sensitive to the lag length p

The ADF-Test: Lag Length Selection Three approaches are commonly used: Akaike Information Criterion (AIC) Schwarz-Bayesian Criterion (SBC) General-to-Specific successive t-tests on lag coefficients AIC and BIC are statistics that favour fit (smaller residuals) but penalize for every additional parameter that needs to be estimated: So, we prefer a model with a smaller value of a criterion statistic General-to-Specific: begin with a general model where p is fairly large, and successively re-estimate with one less lag each time (keeping the sample fixed) It is advised to use AIC Tendency of SBC to select too parsimonious of a model The ADF-test is biased when any autocorrelation remains in the residuals Note: the test critical values do not depend on the method used to select the lag length

Dickey-Fuller (and ADF) Test: Criticism The power of the tests is low if the process is stationary but with a root “close” to 1 (so called “near unit root” process) e.g. the test is poor at rejecting θ = 1 (ψ=0), when the true data generating process is yt = 0.95yt-1 + εt This problem is particularly pronounced in small samples

The Phillips Perron (PP) test Rather popular in the analysis of financial time series The test regression for the PP-tests is PP modifies the test statistic to account for any serial correlation and heteroskedasticity of εt The usual t-statistic in the DF-test … … is modified:

The PP test (2) Under the null hypothesis that ψ = 0, Zt statistic has the same asymptotic distribution as the ADF t-statistic Advantages: PP-test is robust to general forms of heteroskedasticity in εt No need to specify the lag length for the test regression

The Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test The KPSS test is a stationarity test. The H0 is: yt ~I(0) Start with the model: Dt contains deterministic components, εt is I(0) and may be heteroskedastic The test is then H0: against the alternative H1: The KPSS test statistic is: where is a cumulative residual function and is a long-run variance of εt as defined earlier (see slide 32) See Appendix C on some details w.r.t. critical values

Testing for Higher Orders of Integration Just when we thought it is over... Consider: yt = yt-1 + εt we test H0: =0 vs. H1: <0 If H0 is rejected, then yt is stationary What if H0 is not rejected? The series has a unit root, but is that it? No! What if ytI(2)? So we now need to test H0: ytI(2) vs. H1: ytI(1) Regress 2yt on yt-1 (plus lags of 2yt, if necessary) Test H0: ytI(1), which is equivalent to H0: ytI(2) So, if we do not reject, then we conclude yt is at least I(2)...

Working with Non-Stationary Variables Consider a regression model with two variables; there are 4 cases to deal with: Case 1: Both variables are stationary=> classical regression model is valid Case 2: The variables are integrated of different orders=> unbalanced (meaningless) regression Case 3: Both variables are integrated of the same order; regression residuals contain a stochastic trend=> spurious regression Case 4: Both variables are integrated of the same order; the residual series is stationary=> y and x are said to be cointegrated and… You will have more on this in L-5, L-8 and L-9

Cointegration Important implication is that non-stationary time series can be rendered stationary by differencing Now we turn to the case of N>1 (i.e., multiple variables) An alternative approach to achieving stationarity is to form linear combinations of the I(1) series – this is the essence of “cointegration” [Engle and Granger (1987)]

Cointegration Three main implications of cointegration: Existence of cointegration implies a set of dynamic long-run equilibria where the weights used to achieve stationarity are the parameters of the long-run (or equilibrium) relationship. The OLS estimates of the weights converge to their population values at a super-consistent rate of “T” compared to the usual rate of convergence, Modeling a system of cointegrated variables allows for specification of both the long-run and short-run dynamics. The end result is called a “Vector Error Correction Model (VECM)”.

Cointegration We will see that cointegrated systems (VECMs) are special VARS. Specifically, cointegration implies a set of non-linear cross-equation restrictions on the VAR. Easiest/most flexible way to estimate VECM’s is by full-information maximum likelihood.

Long-Run Equilibrium Relationships: Examples Permanent Income Hypothesis (PIH) Postulates a long-run relationship between log real consumption and log real income: Assuming real consumption and income are non-stationary (I(1)) variables, then the PIH is postulating that real consumption and income move together over time and that ut is a stationary series.

Term Structure Of Interest Rates Models the relationship between the yields on bonds of differing maturities. Prior is that yields of different (longer) maturities can be explained in terms of a single (typically shorter) maturity yield. For example: All the yields are assumed to be I(1), but the residuals are I(0) [stationary]. This is an example of a system of three variables with two (2) long-run relationships

VECM Cointegration postulates the existence of long-run equilibrium relationships between non-stationary variables where short-run deviations from equilibrium are stationary. What is the underlying economic model? How do we estimate such a model?

Bivariate VECMs Consider a bivariate model containing two I(1) variables, say Assume the long-run relationship is given by Here represents the long-run equilibrium, and ut represents the short-run deviations from the long-run equilibrium (see next slide).

Phase Diagram: VECM

Adjusting Back To Equilibrium Suppose there is a positive shock in the previous period, raising y1,t to point B while leaving y2,t-1 unchanged. How can the system converge back to its long-run equilibrium? There are three possible trajectories…

Adjustments Are Made by Y1,t Long-run equilibrium is restored by y1,t decreasing toward point A while y2,t remains unchanged at its initial position. Assuming that the short-run change in y1,t are a linear function of the size of the deviation from the LR equilibrium, ut-1, the adjustment in y1,t is given by: where is a parameter to be estimated.

Adjustments Are Made by Y2,t Long-run equilibrium is restored by y2,t increasing toward point C while y1,t remains unchanged after the initial shock. Assuming that the short-run movements in y2,t are a linear function of the size of shock, ut, the adjustment in y2,t is given by: where is a parameter to be estimated.

Adjustments are made by both Y1,t and Y2,t The previous two equations may operate simultaneously with both y1,t and y2,t converging to a point on the long-run equilibrium path such as D. The relative strengths of the two adjustment paths depend on the relative magnitudes of the adjustment parameters, The parameters are known as the “error-correction parameters” or short-run adjustment coefficients.

VECM = Special VAR A VECM is actually a special case of a VAR where the parameters are subject to a set of cross-equation restrictions because all the variables are governed by the same long-run equations. Consider what we have when we put the two equations together: or in terms of a VAR…

VECM = Special VAR

VECM = Special VAR Obviously, we have a first order VAR with two restrictions on the parameters. In an unconstrained VAR of order one, no cross-equation restrictions are imposed, implying 6 unknown parameters. However, a VECM – owing to the cross-equation restrictions – has only four unknown parameters. Less restrictions are needed to identify the model.

Multivariate Methods: N > 2 Can easily generalize the relationship between a VAR and a VECM to N variables and p lags. Assume first that p = 1: Subtracting yt-1 from both sides: or This is a VECM, but with p = 0 lags.

VAR with p lags > 1 Allowing for p lags gives: where vt is an N dimensional vector of iid disturbances and is a p-th order polynomial in the lag operator. The resulting VECM has p-1 lags given by:

Cointegration If the vector time series yt is assumed to be I(1), then yt is cointegrated if there exists an N x r full column rank matrix, , such that the r linear combinations: are I(0). The dimension “r” is called the cointegrating rank and the columns of are called the co-integrating vectors. This implies that (N – r) common trends exist that are I(1).

Granger Representation Theorem Suppose yt, which can be I(1) or I(0), is generated by Three important cases: (a) If has full rank, i.e., r = N, then yt is I(0) (b) If has reduced rank 0 < r < N, then yt is I(1) and is I(0) with cointegrating vectors given by the columns of (c) if has zero rank, r = 0, and yt is I(1) and not cointegrated.

Examples: Rank of Long-Run Models The form of for the two long-run models we considered above: Permanent Income: (N=2, r=1) Term structure: (N = 3, r = 2)

Key Implications of the GE Representation Theorem The Granger-Engle theorem suggests the form of the model that should be estimated given the nature of the data. If has full rank, N, then all the time series must be stationary, and the original VAR should be specified in levels. This is the “unrestricted model”. If has reduced rank, with 0 < r < N, then a VECM should be estimated subject to the restrictions

Key Implications of the GE Representation Theorem If , then the appropriate model is: In other words, if all the variables in yt are I(1) and not cointegrated, we should estimate a VAR(p-1) in first differences. Note that this is the most restricted model compared to the previous two, which is important when calculating likelihood ratio tests for cointegration.

Dealing With Deterministic Components We can easily extend the base VECM to include a deterministic time trend, viz: where now are (N x 1) vectors of parameters associated with the intercept and time trend. The deterministic components can contribute both to the short-run and the long-run components of yt

Deterministic Components Suppose we can decompose these parameters into their short-run and long-run components by defining: where (N x 1) is the short-run component and is the long-run component. We can rewrite the model as:

Deterministic Components The term represents the long-run relationship among the variables. The parameter provides a drift component in the equation of , so it contributes a trend to Similarly allows for linear time trend in and a quadratic trend to By contrast, contributes a constant to the EC-Eq and contributes a linear time trend to EC-Eq

Deterministic Components The equation contains five important special cases summarized on the next slide. Model 1 is the simplest (and most restricted) as there are no deterministic components. Model 2 allows for r intercepts in the long-run equations. Model 3 (most common) allows for constants in both the short-run and the long-run equations – total of N+r intercepts.

Alternative Deterministic Structures

Estimating VECM Models If you are willing to assume that the error term is white noise and N(0,σ2), the parameters of the VECM can be estimated directly by full-information maximum likelihood techniques. Basically, one estimates a traditional VAR subject to the cross-equation restrictions implied by cointegration. Using FIML is the most flexible approach, but it requires one to ensure that the parameters of the overall model are identified (via exclusion restrictions). More on this later.

Three Cases: VECM is equivalent to the unconstrained VAR. No restrictions are imposed on the VAR. Maximum likelihood estimator is obtained by applying OLS to each equation separately. The estimator is applied to the levels of the data, since they are (must be) stationary.

Reduced Rank (Cointegration) Case: FIML If cannot be inverted (i.e., reduced rank case, or we are dealing with a cointegrated system), we impose the cross-equation restrictions coming from the lagged ECM term(s), and then estimate the system using full-information maximum likelihood methods. The VECM is a restricted model compared to the unconstrained VAR.

Reduced Rank Case: Johansen Estimator We can also use the Johansen (1988) estimator. This differs from FIML in that the cross-equation identifying restrictions are NOT imposed on the model before estimation. The Johansen approach estimates a basis for the vector space spanned by the cointegrating vectors, and THEN imposes identification on the coefficients.

Zero-Rank Case for When , the VECM reduces to a VAR in first differences. As with the full-rank model, the maximum likelihood estimator is the ordinary least squares estimator applied to each equation separately. This is the most constrained model compared to a VECM/unconstrained VAR in levels.

Identification The Johansen procedure requires one to normalize the cointegrating vectors so that one of the variables in the equation is regarded as the dependent variable of the long-run relationship. In the bi-variate term structure and the permanent income example, the normalization takes the form of designating one of variables in the system as the dependent variable.

Identification: Triangular Restrictions Suppose there are r long-run relationships. Identification can be achieved by transforming the top (r x r) block of (the long-run parameters) to the identity matrix. If r = 1, this corresponds to normalizing one the coefficients to unity.

Triangular Restrictions If there are N = 3 variables and r = 2 cointegrating equations, one sets to: This form of the normalized estimated co-integrated vector is appropriate for the tri-variate term structure model introduced earlier.

Structural Restrictions Traditional identification methods can also be used with VECM’s, including exclusion restrictions, cross-equation restrictions, and restrictions on the disturbance covariance matrix. Example: Johansen and Juselius(1992) propose an open economy model in which represents, respectively, the spot exchange rate, the domestic price level, the foreign price, the domestic interest rate and the foreign interest rate. Thus, N = 5.

Open Economy Model Assuming r = 2 long-run equations, the following restrictions consisting of normalization, exclusion and cross-equation restrictions on yield the normalized long-run parameter matrix The long-run equations represent PPP and UIP.

Cointegration Rank So far we have taken the rank of the system as given. But how do we decide how many co-integrating vectors are in the vector of N variables? Simple approach is to estimate models of different rank and then do a formal likelihood ratio test to decide whether restricted model (i.e., the model with rank r less than N) is appropriate. Specifically, one would estimate the most restricted model (r = 0), a model that assumes (r=1), then a model that assumes r = 2, etc. The process ends when we cannot reject the null (r = r0).

Cointegration Rank: Likelihood Ratio Test Suppose we estimate the model assuming no cointegration. Let the parameters involved in that model be denoted by Let the value of the likelihood of this model be denoted by Now estimate the model assuming r ≥ 1. Obviously, this is an restricted model compared to the r = N case. Let the value of the likelihood in this case be denoted by

Cointegration Rank: Likelihood Ratio Test Using the standard result for the likelihood ratio test, we get the following LR test statistic: We reject the restricted model if the likelihood ratio test is greater than the corresponding critical value. In this case, imposing the restrictions does not yield a superior model.

Cointegration Rank: Johansen Approach A numerically equivalent approach was proposed by Johansen (1988). He expressed the problem in terms of the eigen values of the likelihood function – an approach that is numerically equivalent to the likelihood ratio test. He termed it the “trace statistic”. The critical values of the LR test are non-standard, and depend on the structure of the deterministic part of the model. Critical values are shown on the next slide.

Critical Values of the Likelihood Ratio Test

Tests on the Cointegrating Vector (Long-Run Parameters) Hypothesis tests on the cointegrating vector, , constitute tests of long-run economic theories. In contrast to the cointegration rank tests, the asymptotic distribution of the Wald, Likelihood Ratio and Lagrange Multiplier tests is under the null hypothesis that the restrictions are valid.

Exogeneity An important feature of a VECM is that all of the variables in the system are endogenous. When the system is out of equilibrium, all the variables interact with each other to move the system back into equilibrium, In a VECM, this process occurs (as we saw) through the impact of lagged variables so that yi,t is affected by the lags of the other variables either through the error correction term, ut-1, or through the lags of

Weak versus Strong Exogeneity If the first channel does not exist, i.e., the lagged error correction term does not influence the adjustment process, the variable concerned is said to be weakly exogenous. If the first and second channels do not exist, then only the lagged values of a variable can be used to explain its changes. In this case, we say that that variable is strongly exogenous. Strong exogeneity testing is equivalent to Granger causality testing.

Example: Exogeneity Consider the bi-variate term structure model with one cointegrating vector. The ten-year interest rate, , is said to be weakly exogenous if Strong exogeneity amounts to the requirement that

Impulse Response Functions The dynamics of a VECM can be investigated using impulse response functions. The approach is to re-express the VECM as a VAR, but preserving the implied restrictions on the parameters. For example, consider the VECM

Impulse Response Functions: VECM This VECM can be expressed as a VAR in levels: subject to the restrictions:

Appendices

Appendix A: Process moments, key results: AR(1) model with θ < 1 Mean (first moment):

Appendix A: Process moments, Simulation of an AR(1) model Assume It follows that Also Note that the sample moments converge to these values as the sample size increases. Also, the variance of the estimator is approaching zero as T increases.

Appendix A: Process moments, key results: AR(1) model with θ = 1 First moment: Second moment: Appropriate scaling factors for these moments are and respectively. Define (sample moments)

Appendix A: Process moments, simulation of an I(1) Process Notice that the variances of the first two sample moments do not fall as the sample size is increased (Columns 2 and 4). The variances converge to 1/3, so m1 and m2 converge to random variables in the limit.

Appendix B: Enders Strategy

Appendix B: Enders Strategy (2) Enders Strategy was criticized for: triple- and double-testing for unit roots unrealistic outcomes: economic variables unlikely contain both stochastic and deterministic trend as in yt = 1+ 2t+  yt-1 + εt , 2≠0,  =0, this possibility should be excluded from the test not taking advantage of prior knowledge Alternative: Elder and Kennedy Strategy

Appendix B: Elder and Kennedy Strategy

Nonstationary Asymptotics

Nonstationary Asymptotics