ARCH and Multiple Time Series

Quantitative Methods 8 min readModule 5 of 5Save

What you'll learn

LOS 2.m
Explain autoregressive conditional heteroskedasticity (ARCH) and describe how ARCH models can be applied to predict the variance of a time
LOS 2.n
Explain how time series variables should be analyzed for nonstationarity and/or cointegration before use in a linear regression.
LOS 2.o
Determine an appropriate time series model to analyze a given investment problem and justify that choice.

series.

In an AR model, autoregressive conditional heteroskedasticity (ARCH) exists if the variance of the residuals in one period is dependent on the variance of the residuals in a previous period. When this condition exists, the standard errors of the regression coefficients in AR models and the hypothesis tests of these coefficients are invalid.

Using ARCH Models

An ARCH model is used to test for autoregressive conditional heteroskedasticity. Within the ARCH framework, an ARCH(1) time series is one for which the variance of the residuals in one period is dependent on (i.e., a function of) the variance of the residuals in the preceding period. To test whether a time series is ARCH(1), the squared residuals from an estimated time series model, $ε^_{t}^{2}$ are regressed on the first lag of the squared residuals $ε^_{t}^{2}$ .

The ARCH(1) regression model is expressed as:

ε^_{t}^{2} = a_{0} + a_{1} ε^_{t - 1}^{2} + μ_{t}

If the coefficient, $a_{1}$ , is statistically different from zero, the time series is ARCH(1).

If a time series model has been determined to contain ARCH errors, regression procedures that correct for heteroskedasticity, such as generalized least squares, must be used in order to develop a predictive model. Otherwise, the standard errors of the model’s coefficients will be incorrect, leading to invalid conclusions.

Predicting the Variance of a Time Series

However, if a time series has ARCH errors, an ARCH model can be used to predict the variance of the residuals in future periods. For example, if the data exhibit an ARCH(1) pattern, the ARCH(1) model can be used in period t to predict the variance of the residuals in period t + 1:

σ^_{t + 1}^{2} = a^_{0} + a^_{1} ε^_{t}^{2}

Worked example: ARCH(1) time series

The next figure contains the results from the regression of an ARCH(1) model. The squared errors for periods t through T are regressed on the squared errors for periods t − 1 through T − 1. ( $μ_{t}$ is the error term for the model.) Determine whether the results indicate autoregressive conditional heteroskedasticity (ARCH), and if so, calculate the predicted variance of the error terms in the next period if the current period squared error is 0.5625.

ARCH (1) Regression Results

Variable	Coefficient	Standard error	t statistic	p value
Constant	5.9068	1.08631	5.4375	< 0.001
Lag 1	0.4515	0.09558	4.7238	< 0.001

Solution

Since the p value for the coefficient on the lagged variable indicates statistical significance, we can conclude that the time series is ARCH(1). As such, the variance of the error term in the next period can be computed as:

σ^_{t + 1}^{2} = 5.9068 + 0.4515 (0.5625) = 6.1608

Occasionally an analyst will run a regression using two time series (i.e., time series utilizing two different variables). For example, using the market model to estimate the equity beta for a stock, an analyst regresses a time series of the stock’s returns ( $y_{t}$ ) on a time series of returns for the market ( $x_{t}$ ):

y_{t} = b_{0} + b_{1} x_{t} + e_{t}

Notice that now we are faced with two different time series ( $y_{t}$ and $x_{t}$ ), either or both of which could be subject to nonstationarity.

To test whether the two time series have unit roots, the analyst first runs separate DF tests with five possible results:

Both time series are covariance stationary.
Only the dependent variable time series is covariance stationary.
Only the independent variable time series is covariance stationary.
Neither time series is covariance stationary and the two series are not cointegrated.
Neither time series is covariance stationary and the two series are cointegrated.

In Scenario 1 the analyst can use linear regression, and the coefficients should be statistically reliable, but regressions in Scenarios 2 and 3 will not be reliable. Whether linear regression can be used in Scenarios 4 and 5 depends upon whether the two time series are cointegrated.

Cointegration

Cointegration means that two time series are economically linked (related to the same macro variables) or follow the same trend and that relationship is not expected to change. If two time series are cointegrated, the error term from regressing one on the other is covariance stationary and the t tests are reliable. This means that Scenario 5 will produce reliable regression estimates, whereas Scenario 4 will not.

To test whether two time series are cointegrated, we regress one variable on the other using the following model:

y_{t} = b_{0} + b_{1} x_{t} + ε

The residuals are tested for a unit root using the Dickey Fuller test with critical t values calculated by Engle and Granger (i.e., the DF EG test). If the test rejects the null hypothesis of a unit root, we say the error terms generated by the two time series are covariance stationary and the two series are cointegrated. If the two series are cointegrated, we can use the regression to model their relationship.

Figure 2.2: Can Linear Regression Be Used to Model the Relationship Between Two Time Series?

Dependent time series	Independent series is covariance stationary	Independent series is not covariance stationary
Covariance stationary	Yes	No
Not covariance stationary	No	Yes, if the two time series are cointegrated

To determine what type of model is best suited to meet your needs, follow these guidelines:

Determine your goal. Are you attempting to model the relationship of a variable to other variables (e.g., cointegrated time series, cross sectional multiple regression)? Are you trying to model the variable over time (e.g., trend model)?
If you have decided on using a time series analysis for an individual variable, plot the values of the variable over time and look for characteristics that would indicate nonstationarity, such as non constant variance (heteroskedasticity), non constant mean, seasonality, or structural change. A structural change is indicated by a significant shift in the plotted data at a point in time that seems to divide the data into two or more distinct patterns. (Figure 2.3 shows a data plot that indicates a structural shift in the time series at Point a.) In this example, you have to run two different models, one incorporating the data before and one after that date, and test whether the time series has actually shifted.

If the time series has shifted significantly, a single time series encompassing the entire period (i.e., both patterns) will likely produce unreliable results.

Figure 2.3: A Structural Shift in a Time Series

If there is no seasonality or structural shift, use a trend model. If the data plot on a straight line with an upward or downward slope, use a linear trend model. If the data plot in a curve, use a log linear trend model.
Run the trend analysis, compute the residuals, and test for serial correlation using the Durbin Watson test.

If you detect no serial correlation, you can use the model. If you detect serial correlation, you must use another model (e.g., AR).

If the data has serial correlation, reexamine the data for stationarity before running an AR model. If it is not stationary, treat the data for use in an AR model as follows:

If the data has a linear trend, first difference the data.

If the data has an exponential trend, first difference the natural log of the data.

If there is a structural shift in the data, run two separate models as discussed previously.

If the data has a seasonal component, incorporate the seasonality in the AR model as discussed in the following.

After first differencing in 5 previously, if the series is covariance stationary, run an AR(1) model and test for serial correlation and seasonality.

If there is no remaining serial correlation, you can use the model.

If you still detect serial correlation, incorporate lagged values of the variable (possibly including one for seasonality e.g., for monthly data, add the 12th lag of the time series) into the AR model until you have removed (i.e., modeled) any serial correlation.

Test for ARCH. Regress the square of the residuals on squares of lagged values of the residuals and test whether the resulting coefficient is significantly different from zero.

If the coefficient is not significantly different from zero, you can use the model.

If the coefficient is significantly different from zero, ARCH is present. Correct using generalized least squares.

If you have developed two statistically reliable models and want to determine which is better at forecasting, calculate their out of sample RMSE.

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