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layout• Unit root testing
• Cointegration
• Vector Auto-regressions
• Cointegration in Multivariate systems
Introduction
• Stationarity or otherwise of a series can strongly influence its behaviourand properties.
• For instance a ‘shock’ dies away with stationarity but is persistent if nonstationary.
• ‘Spurious regressions’ - if variables are trended over time it may producesignificant coefficients and high R2 but it is a meaningless relationship.
• Two types of trend• Stochastic Trend - [random walk]• Deterministic Trend
• Why distinguish between them?
• May look the same but have very different properties
Deterministic trends
Deterministic trends
• Taking the first difference of a trend stationary series removes the non-stationarity but at the cost of introducing an MA(1) process in the residuals.
Non invertible MA process i.e. cannot be written as an AR process
Deterministic trend
Stochastic trends• Consider the following
Stochastic trends • Take this process forward S periods in time:
• As S approaches infinity • the values of Y do not become independent of the error terms; and
• the drift term increases over time
• This process is know as the stochastic trend because it is dependent on the drift and the stochastic progression of error terms
Stochastic trend
Detecting unit root- dickey fuller tests
• Dickey and Fuller (Fuller, 1976; Dickey and Fuller, 1979).- pioneers on testing for a unit root in time series
• The basic objective of the test is to examine the null hypothesis that:
• Against a one sided alternative
Dickey Fuller tests
Reject Null if DF statistic is more negative than the critical values
Augmented Dickey Fuller Unit root Test
Augmented Dickey Fuller Unit root Test • The ADF test requires a specific lag length to augment the
autoregressive process of Yt so as to soak any dynamic structurepresent in the dependent variable and to expunge any possible serialcorrelation in the regression residuals.
• However:• larger lag length may increase the standard errors of the coefficients- degrees
of freedom are used up.
• lower lag length will not remove all the autocorrelation and will bias theestimated results (Enders, 2010; and Brookes, 2010).
• Use information criterion to choose lag length
An Eviews Demonstration
Unit root testing
Augmented Dickey Fuller results
Computed value
Order of integration
• A time series is said to be integrated of order d, I(d), if after differencing d times it becomes stationary
• If a variable is stationary it is said to be I(0)
• If the first difference of a non stationary variable is stationary it is said to be I(1)
• Most economic data is I(1)
Why are we concerned about the order of integration?• Has a direct bearing on the appropriateness and statistical validity of
regression results• If we wish to regress Y on X when:
• Yt and Xt are stationary, classical OLS is valid• Yt and Xt are integrated of different orders, regression is meaningless• Yt and Xt are integrated of same order and residuals are non stationary, regression
may be spurious• Yt and Xt are integrated of same order and residuals are stationary, regression may
indicate a cointegrating relationships
• CLR is founded on asymptotc theory, which implies convergence of variance on a constant
• This is not the case when variables are non stationary- sample moments converge to Brownian Motion or Weiner Processes
Consequences of non stationarity
• Sampling distributions take a non standard form
• We can no longer rely on t and F distributions in statistical inference
• Normal hypothesis testing is invalidated
• There is a tendency to reject the null of no association between individual and all regressors jointly- the proble only intensifies withanincrease in the sample size
High level over-view on Cointegration
Introduction • Modern econometric analysis emphasise the importance of unit root
testing in conducting empirical econometric work.
• Granger and Newbold (1974) non-stationary data yield misleading or spurious regression results i.e. regressions that do not make sense e.g
Introduction • Results exhibit high R2 values which converge to 1, high F and t-
statistics and very low Durbin Watson statistics (serial correlation in residuals.
• Phillips (1986) –a pioneer on asymptotic theory with I(1) variables, concurs with Granger and Newbold and proves that in the above regression:• While beta parameters should converge to zero as sample size approaches
infinity, they are non zero;
• R2 statistics approach ; and
• T-statistics approach infinity.
• Brooks (2010), results reflect contemporaneously correlated time trends instead of the true underlying relationships
Introduction
• To avoid spurious regressions and to compute BLUE parameters variables are often differenced to achieve stationarity
• However, economic and finance theory is anchored on long run relationship and differencing removes long run information from the data series
Cointegration • Engle and Granger (1987) – it is possible to estimate valid regressions using
non-stationary data.
• Develop a technique to estimate valid parameters and to test for longrunrelationships between nonstationary variable (Granger Representation Theorem)
• A set of non-stationary variables integrated of the same order, say I(1), are linked to form an equilibrium relationship spanning the long-run if they combine to form a lower order series integrated of the order I(0), they are said to be cointegrated.
Cointegration
• This implies that variables will moveclosely together and will not driftarbitrarily over time and thedistance between them will bestationary
• The concept of cointegration mimicsthe existence of a long-runequilibrium relationship to whichthe variables converge over time.
• the distance that the system is awayfrom equilibrium at any given time istermed the equilibrium error
Cointegration
• The distance that the system is away from equilibrium at any given time is termed the equilibrium error.
• It allows for a richer study of the short-run dynamics of adjustment towards equilibrium through the use of error correction models.
Cointegration in bivariate systems
Testing for Cointegration (residuals based test)
Cointegration and error correction
Procedure in testing for Cointegration
Two step Engel and Granger procedure
• Step 1: Run a static regression in levels between the variables
• Save the residuals series: and
• Step 2: Test for stationary of residuals• If stationary- Cointegration, proceed to estimate ECM
• If non stationary- No Cointegration
Step 1:Estimating a static Longrun equation
Go to Quick, then select estimateEquation on the drop down menu
Step 1:Estimating a static long run equation
In the equation dialog box Type the equation you wish to estimate. Always remember to include a constant
Step 1: long run equation resultsRecall spurious regressions
Caution
• Check whether the coefficients in the long-run equation conform to apriori expectations in terms of direction of the impact and not on the magnitude and significance of the coefficients.
• The Rsqr statistic is useless and should not be interpreted.
Step 1: Creating a residual series/ equilibrium error
To create a residuals series Go to the proc button and select the Make residual series option
Create residual series/ Equilibrium error
Creating residual series/ Equilibrium error
Name the residual series and Click OK
Testing for unit roots in the residuals
Once the residual series is createdClick on view and select the unit root testbutton
Step 2:Testing for Cointegration
Select the ADF test on the testType window
Select the level button
Always select nonefor the ADF model when conducting unit root tests
Step 2 residual test
Theoretically, the ADF critical values are not valid
Should ordinarily be based on MacKinnon surface response functions, Harris, 1995.
However, in practiceADF is used as a proxy for the true critical values
Since the ADF is more negative than the critical values we reject the null that the variables are not cointegrated
Step 2: Estimating an error correction model
• The error correction model also known as the dynamics ofadjustment are estimated using the lagged differences of the dataseries and the lag of the equilibrium error we have calculated above
Estimating an error correction model
In the equation dialogue box enter the variables in their differences
First difference of lm3 i.e. money supply
constant Lagged equilibrium error
Estimate an error correction (2 step model)
Valid regressions?
Error correction term/ Speed of adjustment
Estimating a one step EG Cointegration ECM equation• After testing for Cointegration as above, proceed to estimate an error
correcting model
• In the equation dialogue box type the following equation
Lm3(-1) Captures the speed of adjustment towards equilibrium
Longrun componentShortrun dynamics
Cointegration- One step Engel Granger Procedure
Error correction term/ speed of adjustment should always be negative
Calculation of Elasticities
• The elasticity of money demand to the changes in the longru variables are calculated as follows: e.g. the elasticity of nominal gdp
• LNGDP elasticity=(coefficient of gdp)/(coefficient of adjustment
= 0.1575/0.0875
=1.8
• Therefore we would say that a 10% change in nominal income will result in an 18% change in the money demanded in the longrun.
• Note that: the coeffient in the longrun equation estimated in the two step procedure and the elasticity above are almost the same
• This is due to the superconsistency property of OLS
Diagnostic tests- subject equations to a battery of tests
Whilst the equation is still open, click on View to see the menu of diagnostic tests
Diagnostic testing (plot of residual series)
Correlogram of residuals
Correlogram of squared residuals
Residual tests- normality test
Serial correlation test Fail to reject the null of no serial correlation
Heteroskedasticity tests Fail to reject the null of no Heteroskedasticityat 5%
Cumulative sum of residuals test
Cumulative sum of squared residuals tests
Equation is unstable
Investigate and find out why..In this case it’s the run-up to the attainment of independence and the end of apartheid.. To control for this we need dummy variables
Multivariate Cointegrationand Vector Auto-regressions
Advantages of the Engle and Granger approach• Relatively simple
• Useful as a first indication of the existence of a longrun equilibrium relationship
• Where there is a consistent Cointegration vector it allows us to use the superconsistency property of OLS to obtain consistent estimates of the cointegrating vector
• Provides longrun equilibrium information and the short term dynamics
• Provides speed of adjustment to equilibrium
Limitations of the E_G approach• Distribution of test statistics is only a rough guide and will be slightly
different in any application
• For more than two variables it is no longer possible to demonstrate the uniqueness of the Cointegration vector• If we have a vector of N variables each integrated of the same order, we can have
up to N-1 Cointegration vectors
• Has no systematic procedure to estimate multiple Cointegration vectors
• Results are based on asymptotic theory but we do not have infinitely large samples in practise
• Carry over error bias
Multivariate cointegration
• Johansen and Jesilius (1988) and Stock and Watson (1988) develop max likelihood procedure to test for Cointegration
• Their test could estimate and test the number of cointegrationequations and to test restricted versions of the cointegrating vectors and speeds of adjustment
• Allows verification of theories through coefficient restrictions e.t.c
• The test based on the stationary VAR
Vector Autoregressive (VAR) models• We popularised by Sims(1980) as a natural generalisation of
univariate autoregressive models.
• Variables should be treated symmetrically to avoid ‘incredible identification restrictions’
• Let the data speak for itself i.e. no apriori assumption about exogeneity of variables
• Very helpful in identifying the relationship among a set of macroeconomic models
Vector Autoregressive (VAR) models
• Multiequation time series model
• Considers a number of interrelated variables
• Imposes zero restrictions on estimation of parameters
• Atheoretical i.e. no strict reliance on theory to formulate the model
• ‘Everything causes everything’
• However, the number of estimated parameters makes the model difficult to interpret
Vector Autoregressive (VAR) models
• Advantages of VARs over simple regression models:• Every variable is endogenous (no incredible exogeneity assumptions).
• Every variable depends on the others (no incredible exclusion restrictions).
• Simple to estimate and use.
• General disadvantages:• It is a reduced form model; no economic interpretation of the dynamics is
possible.
• Potentially difficult to relate VAR dynamics with DSGE dynamics (which have an ARMA structure)Can be specified as follows
• Can’t be used for certain policy analyses (Lucas critique).
Vector Autoregressive (VAR) models
• Multi-equation time series model
𝑌𝑡 = 𝜇 + 𝑖=1𝑘 θ𝑖 𝑌𝑡−𝑖 + 𝜀𝑡
Yt is a (𝑚 × 1) vector of I(0) variables
𝜇 is a (𝑚 × 1) vector of constants, and
θ1… . . 𝜃𝑘are (𝑚 ×𝑚) matrices of parameters,
k is the appropriate lag length of the model,
𝜀𝑡 is a ( 𝑚 × 1) vector of normally distributed error terms.
Vector Autoregressive (VAR) models
• The properties are:• The variabels are stationary
• Error terms are white noise disturbances with a constant variance
• Error terms are not serially correlated
• The structure of the system allows for feedback effects
• If contemporanoues effects are assumed to be zero the VAR is said to be in standard form and estimation can proceed using OLS
Vector Autoregressive (VAR) modelsStationarity and VAR
• Brookes (2010) it is important that all of the variables in the VAR processbe stationary otherwise hypothesis are invalid
• Sims (1980) and Sims, Stock and Watson (1990) as cited in Enders (2010)recommend against differencing even if variables contain a unit root.
• Argue that the objective of VAR analysis is to determine interrelationshipsamong variables and not to determine parameter estimates.
• Also argue against detrending data in a VAR
• Canova (2005) If we want a constant coefficient VAR, we needstationarity of the variables. If non-stationarities are present a VARrepresentation exists, but with time varying coefficients.
Vector Autoregressive (VAR) models
• To determine the appropriate lag length/ order of VAR• Akaike information criterion• Schwarts information criterion • Likelihood ratio test• Final prediction error• HQ information criterion• Maximum lag for autoregressive models• Experimentation i.e. general to specific modeling
• You choose the lag length that soaks or expunges serial correlation in the residuals
• If maximum lag length is p, then it’s a VAR(p)
Example of a VAR
Variance decompositions
• Enders (2010) enables us to study the variation in Y that is due to its own shocks versus the component of the variation that is due to shocks in other variables
• help determine the relative importance of each innovation in explaining the variables in the system.
• To conduct variance decompositions, the AR process is inverted into an MA process of the errors using Walds Decomposition Theorem
• Rewrites the AR process into
𝑌𝑡 = 𝜇 +
𝑖=1
𝑘
θ𝑖 𝑌𝑡−𝑖 + 𝜀𝑡 𝑋𝑡 = 𝜇 +
𝑖=0
∞
𝜃𝑖𝜀𝑡−𝑖
Forecast error variance decomposition
In a 2 variable case
Forecast error
This reduces to Variance in Y due to itself
Due to others
Variance decomposition• If the forecast error variance is explained by shocks in the variable itself, the the
variable is exogenous
• It is typical for a variable to explain almost all its forecast error variance for short horizons and smaller proportions at longer horizons (Enders, 2010)
• It is also subject to an under identification problem as is the impulse response function, thus there might be need to place additional restrictions on the system in order to obtain the decomposition and impulse responses
• One such restriction is the Choleski decomposition• The contemporaneous value of Y has no contemporaneous effect on X
• This implies an ordering of the variables
• Brooks and Tsolacos (1998) and Enders (2010- the Choleski ordering of the variables has important ramifications on the resulting impulse responses and variance decompositions and is equivalent to an identifying restriction on the VAR.
Impulse response functions
• Impulse responses allows for tracing the time profile of various shocks on the variables in the VAR system
Impulse response function
• Impulse response functions are a practical tool which aid in visualising the behaviour of the variables understudy in response to various shocks.
• They show the dynamics of transmission of shocks, direction and magnitude of the shocks.
• In practice you should always plot your impulse responses together with their standard deviation bands
Multivariate Cointegration
• Johansen and Jesilius enhance the VAR (p) by including the long runcomponents (cointegrating relations) in the VAR (p) process i.e.separating permanent effects from transitory effects.
• It specifies a VECM among variables
Multivariate Cointegration
The cointegrated VAR and VECM
Multivariate cointegration
Testing for Cointegration
• JJ suggest five assumption on which test can be conducted
1. No deterministic trends in the VAR system and the cointegratingrelationship has no intercept and no trend;
2. No deterministic trends in the VAR system and the cointegratingrelationship has an intercept and no trend;
3. Linear trend in the VAR system and the cointegrating relationship has no trend but has an intercept;
4. Linear trend in the VAR system and the cointegrating relationship only has a deterministic trend; and
5. A quadratic trend in the VAR and the cointegrating relationship has a linear deterministic trend.
Testing for cointegration
Testing for cointegration
• Trace statistics tests the null hypothesis that the rank r = 0 (i.e. no cointegration) against the alternative that r > 0 (i.e. there is one or more cointegrating vectors).
• The maximum Eigenvalue statistics on the other hand tests the null hypothesis that the number of cointegrating vectors is r against the specific alternative of r + 1 cointegrating vectors
Testing for Cointegration rank
• The critical values for the tests are obtained using Monte Carlo approach
• The distribution of statistics depends on two components:• The number of non stationary components under the null hypothesis
• The form of the deterministic components, constant, trend or both- has similarity with the Dickey fuller test
• Sometimes the two tests may give conflicting results
• Harris(1995) the maximum eigen value has a sharper alternative hypothesis and is preferred to pin down the number of cointegratingvectors.
• The sequence of the Trace tests leads to a consistent procedure.
Testing for cointegration rank
• Cheung and Lai (1993) propose choosing cointegration rank based on the Trace statistic. They state that the trace statistic is more robust to skewness and excess kurtosis in residuals than the maximum Eigen value statistic.
• Enders (2010) concurs with these findings and states that when the two tests for cointegration rank are in conflict the Trace statistic is likely to give more reliable results.
• Current practise is to only consider the Trace test.
Practical demonstration of Multivariate Cointegration in E-views
Step 1: Pretest data• Pretest all variables to determine their order of integration i.e. test
for unit roots
• Plot the variables to see if a linear time trend is likely to appear in the data series
Step 2: Estimating an Unrestricted VAR
Go to Quick and select estimate VAR
Estimating a VAR
Enter the variables of interest ad click OK
Choose the sample size over which to estimate the VAR
Step 3: Choosing the optimal lag length
In the estimated VAR window, go to View, Lag structure, lag length criteria
Choosing the optimal lag length
Leave the default and click ok
Selecting the lag length
If a long lag is required to ake residuals white noise, reconsider the choice of variables and look for another important explanatory variable to include in the information set
Asterisk indicates lag length selected by Information criteria
Selecting the lag length
• A summary of test statistics that measures the magnitude of the residual autocorrelation in given b the Portmanteau test
• Eviews uses Wald Lag exclusion tests to determine the default lag
• In our case we will select the lag based on SIC (a more stricter test)• AIC gives a generous lag lenth
• HQ is a middle of the road approach
• Chosen lag length for this exercise is 2
Step 4: Deterministic trend specification of the VAR• The variables may have non zero means and deterministic and
stochastic trends
• Similarly Cointegration equations may have intercepts and deterministic trends
• Since the asymptotic distributions of the LR test statistic for Cointegration does not have the usual Chi Square distribution and depends on the restrictions we make with respect to deterministic trends, we need to make assumptions regarding trends underling our data.
• Eviews allows for the 5 trend specification of Johansen and Jesilius
Step 5: Estimation and Determination of Rank
Go to view and select Cointegration tests
Step 5: Estimation and Determination of Rank
Choose the trend assumption you have made in 4: in unique circumstances will you consider a trend in the Cointegration vector
Enter the chosen lag length from step 3 and click OK
Step 5: Estimation and determination of rank
The test is done in specific order from the largest eigen value to the smallest.
We use the ‘Pantula Principle’ where we test for significance until you no longer reject the null
The first null is that there is non stationary relations in the data (r=0)…
So long as TS/MES> critical value reject the null
Use p =-values to make decision
Step 5: Estimation and determination of rank
Click estimate, Select Vector Error Correction
Cross check if lag interval is correct and click on the Cointegration tab
Step 5: Estimation and determination of rank
Enter the number of Cointegration equationsand click OK
Estimations
Cointegration equations/ Longruncomponent
Speed of adjustment
The 2 separate longrunrelationships enter into each of the 4 equations
Estimation
• NB: even if we are only interested in the first cointegratingrelationship, both coinntergation relationships should enter that equation separately.
• There are in effect two ECM terms in the equation
• There is a longrun positive relationship between money supply and inflation and a negative relationship betwee n money supply and the interest rate
• And approximately 8% of deviations in the money supply from its long run equilibrium are cleared in the next quarter
Step 6: Diagnostic testing
• Once a VEC is estimated, a number of diagnostic tests should be performed
• These tests assist in checking the appropriateness of the estimated VAR
• Residual tests:• Portmanteau Autocorrelation test: computes the multivariate Box-Pierce/
Ljung Box Q statistics for serial correlation upto a specified order. Eviewsreports both tests under the null hypothesis of no serial correlation
• Autocorrelation LM test: reports the multivariate LM test statistics for residual serial correlation. Under the null hypothesis of no serial correlation of order h the LM test is asymptotically Chi Square distributed with K^2 degrees of freedom
Step 6: Diagnostic testing
To conduct residual tests go to view, residual tests, portmanteau test
Residual tests
Reject the null of probabilitys of the Q-stat and Ajd Q are less than 0.05
Diagnostics residual tests
We reject the Null Hypothesis of lag order 2 in the residuals of the VECM and conclude that residuals are autocorrelated
Normality tests
• Reports the Multivariate extension fo the Jarque Bera test
• For the multivariate test you must choose a factorisation of residuals that are orthogonal to each other:• Cholesky
• Inverse square root of residual correlation matrix Doornik and Hansen (1994)
• Inverse square root of residual covariance matrix Urza (1997)
• Factorisation from identified VECM
• Eviews reports the test statistics for each othorgonal component
Normality
Choose factorisation method, select cholesky
Normality
The P is the inverse of the lower triangular Cholesky factor of the residual covariance matrix
Reports the joint normality test for our four component equation
Normality
• NB: Paruolo (1997) points out that if normality of the error terms is rejected for other reasons (kurtosis), Johansen results are not affected.
• That is we should not worry if our skewness results are fine.
Whites heteroskedasticity test
• Is an extension of the white’s 1980 test • No cross terms: uses only levels and squares of regressors
• Cross terms: includes all non redundant cross products of regressors(heteroskedasticity of an unknown form)
Heteroskedasticity
Reject the null of no heteroskedasticity
Impulse responses
Click on Impulse, then in the impulse box select the variables you wish to shock and in the responses box select the variable you want to be affected Then click on impulse definition
Choose the number of quarter for the graph
Cholesky ordering
Select the order of the variables i.e. as identified in theory …
This is similar to an identification restriction on the impulses
Impulse responses
Profile and direction of shocks
Variance decompositions
Click on view then choose variance decomposition
Variance decomposition
Select the table option and specify the choleskyordering
Variance decomposition
LNGDP accounts for about 28% of objservedvariations in money supply
Testing theoretical restrictions
Click estimate, the VECM restrictions, impose restrictions
What can go wrong in Johansen Methodology
• We need normally distributed white noise
• The test is asymptotic and can be sensitive to how we formulate the VECM model in limited samples
• Test assumes there are no structural breaks
• If we put a stationary variable in the model, the number of cointegrating vectors may increase
• Weak exogeneity: if weak exogeneity is foun then use a single equation model
References
• Enders, W., 2010, Applied Econometric Time Series 3e, Wiley, USA
• Brookes
