Modeling non-stationary time series

Universidad Privada Boliviana / Universidad del Pacífico (Lima, Perú)

Prof. J. Dávalos (Ph.D.)

Error correction representation of cointegrated series

Cointegration examples

Consumption function

Total consumption = permanent + transitory, both I(1). Given the long run share between permanent income (\(y_p\)) and permanent consumption(\(c_p\)) \(\beta\):

\(C_{total} = \beta y_p + c_{trans}\)

where \(C_{total}\), \(\beta y_p\) are expected I(1) while \(c_{trans}\) is I(0).

Forward vs spot rates

  • Accurate pricing expectations imply that the futures market is a good predictor of the current (spot) price:

  • \(f(t+1)_t = s_{t+1} + \varepsilon_{t+1}\)

  • Today’s (t) forward price for (t+1) is on average equal to the observed (spot) price on t+1. Both prices are well known not to be I(0). Most likely I(1).

  • If this is true, taking advantage of the forward price will not lead to systematic gains. If it does not hold, speculating may be a profitable strategy.

PPP arbitrage

  • Consider the Big Mac Index

  • Under the one price law, the price of a given good or service in two different markets (currencies) should be equal on average. Short-term deviations (\(\varepsilon\)) may lead to arbitrage and price equalization.

  • On average the foreign price \(p^*\) should match the local price \(p\) given the nominal exchange rate between the foreign and local prices (\(e\)): \(p^*= e \;p\)

  • Write the previous equation in log and add a non-systematic stationary discrepancy between both terms:

Exercise 1

Simulate a cointegrated process for \(X_t\) and \(Z_t\) given t=100 (you decide on remaining simulation parameters). Analyse the correlogram of the estimated residuals from an OLS regression between these variables.

Error correction representation

  • We know how to test for the presence of cointegration using the DF or ADF test.
    • Residuals were stationary I(0). This guarantees cointegration and the existence of an error correction representation
    • However cleaner residuals are required to implement statistical inference with standard estimators (OLS, ML)
    • Thus, after finding cointegrated variables, the econometrician may introduce additional k lags of the many variables to get WN errors. This distributes additional lags to the model.

Autoregressive Distributed Lag (ADL) formulation:

  • \(X_t = \alpha_0 +\alpha Z_t + \sum_{j=1}^k \theta_j Z_{t-j} + \sum_{j=1}^k \gamma_j X_{t-j} + \varepsilon_t \quad \quad(1)\)
    • First differentiating, an ADL as well…
  • \(\Delta X_t = \alpha \Delta Z_t + \sum_{j=1}^k \theta_j \Delta Z_{t-j} + \sum_{j=1}^k \gamma_j \Delta X_{t-j} + \varepsilon_t -\varepsilon_{t-1}\)
    • where \(\varepsilon_{t-1}\) is given by (1):
  • \(\Delta X_t = \alpha \Delta Z_t + \sum_{j=1}^k \theta_j \Delta Z_{t-j} + \sum_{j=1}^k \gamma_j \Delta X_{t-j} + \varepsilon_t - \Big( X_t - (\alpha_0 +\alpha Z_t + \sum_{j=1}^k \theta_j Z_{t-j} + \sum_{j=1}^k \gamma_j X_{t-j}) \Big)\)

Error correction model representation

  • Arranging terms :

  • \(\Delta X_t = \alpha \Delta Z_t + \sum_{j=1}^k \theta_j \Delta Z_{t-j} + \sum_{j=1}^k \gamma_j \Delta X_{t-j} + \eta \,\textrm{ec}_{t-1} +\varepsilon_t\)

  • The ADL includes an error correction term \(\textrm{ec}_{t-1}\) whose coefficient (\(\eta\))is expected negative:

    • \(\textrm{ec}_t = X_t - (\alpha_0 +\alpha Z_t + \sum_{j=1}^k \theta_j Z_{t-j} + \sum_{j=1}^k \gamma_j X_{t-j})\)

  • This is interpreted as a deviation from the long-run residual average (0). A positive deviation in a given period means that X is larger than its forecast, thus leading to a reduction in X during the forthcoming period. It may be interpreted as an empirical economic equilibrium condition that self-corrects itself.

  • The \(\textrm{ec}_t\) represents the long-run relationship or cointegrating relationship , aka cointegrating vector.
  • Note however, that this is not unique, as it can be represented for \(Z_t\) instead of \(X_t\).

Alternative generalization

  • The previous assumes that there is one reference variable that dictates the cointegrating relationship. This may not be true. There might be many, none, etc.
  • Thus, the ECM could be written using alternative reference variables. We need a more general approach to identify the cointegrating relationships and ECM.
  • Consider \(\bf X_t\) as a vector of \(n\) I(1) endogenous variables included in the relationship.
  • Also, let’s write the system as we did with the Dickey Fuller representation in levels.
    • \(\bf X_t = A \bf X_{t-1} + \bf\varepsilon_t\) (This is quite general, it allows the presence of unit-root or not like in the ADF)
  • Substracting the first lag just like in the univariate DF test:
    • \(\bf X_t -\bf X_{t-1} = A \bf X_{t-1} -\bf X_{t-1}+ \bf\varepsilon_t\)
    • \(\Delta \bf Xt = \pi \bf X_{t-1} + \bf \varepsilon_{t}\)
      • This is clever. The \(\pi\) matrix informs whether the levels are informative to the FD

EC terms from \(\pi\)

  • Whiteboard…
  • If \(\pi\) does not exist i.e. its rank = 0, then, there are no cointegration relationships and the vector \(\bf X_t\) contains purely stationary variables.
  • The rank of \(\pi\) dictates whether there is cointegration and the number of cointegration vectors and ECM’s.
  • Had we need more lags to get WN errors does not change anything:
    • \(\Delta \bf X_t = \pi \bf X_{t-1} + \sum_{j=1}^k \bf \Delta X_{t-j}+ \bf \varepsilon_{t}\)

Exercise 2

  • From the previous condition, show that 3 error correction terms can be obtained from \(\pi\) for n = 3

Johansen test

Consider the general notation (STATA) for a test on the levels (Y):

One may specify several alternative to a test on the rank of \(\pi\)

Example

  • Cointegration relationship between personal income, investment and consumption for the US.
  • The most standard configuration of I(1) is to drop the linear deterministic trend in first differences \(\tau = 0\). Including such trend may be informative for series with quadratic trends in levels.
  • The second most usual option is not to assume the presence of deterministic trend in the cointegrating relationship, thus we need \(\rho = 0\). A priori acknowledge of the variables of interest may point to stochastic trend variables (stock prices, interest rates, etc.). Assuming \(\tau=\rho=0\) is the default by stata.
  • Choice of lags.
    • The varsoc command and its Likelihood ratio test, provide good insights. Report your test from alternative lags in the neighborhood of the suggestion given by the test.
  • Null hyphotesis - maximum rank test.
    • Each line of the rank test provides a statistic to the test for a maximum number of cointegration vectors. If there is no cointegration, one should not reject the null at the very first line when max_rank=0.
    • Under cointegration, you are expected to reject the first rank =0, and to move forward to test for rank =1, etc. One stops once the null is not rejected.
    • If you always reject the null, the test might be misspecified. Check your lags and options.
webuse balance2
varsoc y i c
vecrank y i c, lags(3) // it always rejects the null, misspecification problem
vecrank y i c, lags(4) // As suggested by the LR test, it always rejects the null
vecrank y i c, lags(5) // finally stops rejecting the null at r=2
vecrank y i c, lags(6) // finally stops rejecting the null at r=2
(macro data for VECM/balance study)


Lag-order selection criteria

   Sample: 1960q1 thru 1982q4                               Number of obs = 92
  +---------------------------------------------------------------------------+
  | Lag |    LL      LR      df    p     FPE       AIC      HQIC      SBIC    |
  |-----+---------------------------------------------------------------------|
  |   0 |  494.087                     4.6e-09  -10.6758  -10.6426  -10.5936  |
  |   1 |  1178.08    1368    9  0.000 2.0e-15  -25.3496  -25.2169  -25.0207  |
  |   2 |  1235.62  115.08    9  0.000 6.8e-16  -26.4049  -26.1725  -25.8292* |
  |   3 |  1251.51  31.766    9  0.000 5.9e-16* -26.5545* -26.2226* -25.7322  |
  |   4 |  1260.27  17.518*   9  0.041 6.0e-16  -26.5493  -26.1178  -25.4802  |
  +---------------------------------------------------------------------------+
   * optimal lag
   Endogenous: y i c
    Exogenous: _cons


Johansen tests for cointegration
Trend: Constant                            Number of obs  = 93
Sample: 1959q4 thru 1982q4                 Number of lags =  3
--------------------------------------------------------------
                                                      Critical
Maximum                                        Trace     value
   rank  Params           LL  Eigenvalue   statistic        5%
      0      21    1245.9796           .     41.2274     29.68
      1      26    1256.8257     0.20804     19.5353     15.41
      2      29    1261.8713     0.10283      9.4441      3.76
      3      30    1266.5933     0.09656
--------------------------------------------------------------


Johansen tests for cointegration
Trend: Constant                            Number of obs  = 92
Sample: 1960q1 thru 1982q4                 Number of lags =  4
--------------------------------------------------------------
                                                      Critical
Maximum                                        Trace     value
   rank  Params           LL  Eigenvalue   statistic        5%
      0      30    1236.9203           .     46.6906     29.68
      1      35    1250.8896     0.26190     18.7520     15.41
      2      38    1256.7437     0.11950      7.0439      3.76
      3      39    1260.2657     0.07371
--------------------------------------------------------------


Johansen tests for cointegration
Trend: Constant                            Number of obs  = 91
Sample: 1960q2 thru 1982q4                 Number of lags =  5
--------------------------------------------------------------
                                                      Critical
Maximum                                        Trace     value
   rank  Params           LL  Eigenvalue   statistic        5%
      0      39    1231.1041           .     46.1492     29.68
      1      44    1245.3882     0.26943     17.5810     15.41
      2      47    1252.5055     0.14480      3.3465*     3.76
      3      48    1254.1787     0.03611
--------------------------------------------------------------
* selected rank


Johansen tests for cointegration
Trend: Constant                            Number of obs  = 90
Sample: 1960q3 thru 1982q4                 Number of lags =  6
--------------------------------------------------------------
                                                      Critical
Maximum                                        Trace     value
   rank  Params           LL  Eigenvalue   statistic        5%
      0      48    1222.3232           .     48.3578     29.68
      1      53    1238.0479     0.29492     16.9084     15.41
      2      56    1244.9553     0.14230      3.0936*     3.76
      3      57    1246.5021     0.03379
--------------------------------------------------------------
* selected rank

There is evidence of 2 cointegrating relationships, thus 2 error correction model representation.