Appendix 6.
Time Series Tests

(c) 2017 by Barton Paul Levenson

To find the proper lag length for integration tests:

SSE AIC = 2 k + N ln (———) N

where k is the number of parameters in the model, SSE is the sum of squared errors (e.g. from a regression or analysis of variance), and N the number of points. Minimize AIC to find the correct lag length.

Note: Many researchers feel the AIC is too generous, keeping variables which should really be excluded. It is better to use the "corrected AIC" AICc:

SSE N + k ln(———) + ————— N N - k - 2

SSE SIC = p ln N - 2 ln (———) N

To find the level of integration of a variable:

This one is complicated. You have to run three different regressions:

Δy_t = γ y_t-1 + δ₁ Δy_t-1 ... + δ_p Δy_t-p + ε_t Δy_t = α + γ y_t-1 + δ₁ Δy_t-1 ... + δ_p Δy_t-p + ε_t Δy_t = α + β t + γ y_t-1 + δ₁ Δy_t-1 ... + δ_p Δy_t-p + ε_t

The dependent variable here is the change in your original variable with time:

Δy_t = y_t - y_t-1

And you're regressing it, first, on the previous value of Y, without an intercept. To do a linear regression with no intercept, one "forced through the origin," you use a column of 0s in your X matrix instead of 1s.

Then you do the regression with an intercept, known in this test as a "drift term," then with an intercept and a time term, known in this test as a "trend term."

What makes it an "Augmented" Dickey-Fuller test, as opposed to just a Dickey-Fuller test, is the presence of the past values of your dependent variable:

δ₁ Δy_t-1 ... + δ_p Δy_t-p

Here p is the appropriate lag, determined by an information criterion such as the SBIC, or one of many other possible arcane methods.

The t-statistics of interest are those on the lagged original-variable term, y_t-1. The t statistic for the coefficient, γ, is called τ_NC for the first regression, τ_C for the second, and τ_CT for the third. The subscripts stand for "No Constant," "Constant," and "Constant plus Trend," respectively.

Unfortunately, these τ ("tau") statistics are not like normal t statistics. You have to get them from special tables, which are hard to find on-line but can be found in some statistics textbooks. Statistics packages which do the ADF tests automatically will calculate the p values for you. The R-language-based freeware package Gretl is an especially good one.

If at least one of the p values from these regressions is very small, your time series is not integrated; i.e., is I(0), or stationary. If none of them are, it's integrated. Repeat the ADF test with the next level of differencing to see it it's merely I(1). And so on.

To find cointegration in a regression involving one or more integrated variables:

This test is in two parts.

1. Test for unit roots (ADF test) in each variable, using a test such as the ADF test (there are others).
2. Draw up a "cointegrating regression," and test for a unit root in the residuals from that.

If we

 A) can't reject unit roots for the variables, but

 B) can reject them for the residuals of the cointegrating regression,

Then a "cointegrating relationship" exists, and your cointegrating regression is valid.

To test for Granger causality:

The Sims Test (Granger 1969, Sims 1972):

First, regress Y on both lagged Y and lagged (not current) X:

_{p p} Y = Σ b_i Y_t-i + Σ c_i X_t-i + u_t ^{i=1 i=1} H₀: c_i = 0

Then on lagged Y alone:

_p Y = Σ b_i Y_t-i + e_t ⁱ⁼¹ H₀: b_i = 0

Compute SSE for each:

_T SSE_XandY = Σ u_t^{2 t=1} _T SSE_Xonly = Σ e_t^{2 t=1}

Finally, compute this statistic:

(SSE_Xonly - SSE_XandY) / p S = —————————————— SSE_XandY / (T - 2p - 1)

for F_p,T-2p-1.

To avoid asymptotic effects,

T (SSE_Xonly - SSE_XandY) S2 = ————————————— SSE_XandY

for Χ(p).

Note: This test is very sensitive to choice of lag length and stationarity issues.