Jacob: Do we solve for the ordinary least squares estimators by the equations in the textbook?

Rachel: Excel provides a built-in regression function. If we use a hundred simulation to test the accuracy of the estimators, we code a VBA macro or an Excel table with the formulas from the textbook. For the student project, it is easier to use the regression built-in function.

Jacob: Where is the Excel regression built-in function?

Rachel: Choose the tools menu from the menu bar. From the menu, choose data analysis. You may have to include the data analysis add-in to your version of Excel. From data analysis, choose regression.

Jacob: Simpler Excel built-in functions determine a linear trend line using regression. Can we use those built-in functions?

Rachel: Those Excel functions use a single independent variable. For two or more independent variables, we need the Excel add-in.

Jacob: How do we include the data analysis add-in?

Rachel: Check to see if the add-in is already installed. Some actuarial departments have the add-in installed.

If the add-in is not installed, choose add-ins… from the tools menu. From the menu that appears, choose analysis toolpak. To work with VBA, include also the analysis toolpak vba.

Your version of Excel may differ. If you can’t find the regression built-in function, post a question on the discussion forum, listing your version of Excel and of windows.

Rachel: If A = 1, the process is not stationary. It is a random walk, and we take first differences.

Jacob: If A = 1 and B = 0, isn’t the current value also the mean?

Jacob: Can we use the same data as on the NEAS spreadsheets? Or should we use other data?

Rachel: You can use the same data or other data; the choice is yours.

Jacob: Why not place more data on the web site?

Rachel: We placed limited data on the discussion forum because some candidates fear they have missed something if they have not used all the data.

Jacob: If we use the same data, won’t we get the same results?

Rachel: The sample workbook shows illustrations, not results. Even for the eras of each series, you might choose other periods. Your project might focus on the choice of eras, examining means, drifts, and variances in various periods, fitting models, and making forecasts.

Jacob: Would comparing pre-9/11 vs post-9/11 interest rates be a good project?

Rachel: This is a perfectly good project, though it may not show much. You would compare the two eras and see if any differences are significant.]

Jacob: Should we examine six month seasonality or 12 month seasonality?

Rachel: We normally examine 12 month seasonality, unless we have a reason to assume six month seasonality.

Jacob: What if the six month sample autocorrelation is significant?

Rachel: Annual seasonality may cause a six month autocorrelation. Use a 12 month AR parameter, and see if the six month sample autocorrelation is still significant. If it is, compare out-of-sample forecasts for six month seasonality vs 12 month seasonality. This is a potential student project.

Jacob: What might cause six month seasonality?

Rachel: Auto premiums and policies have six month seasonality because of the six month policies. Actuarial exam statistics have six month seasonality because of the semi-annual exam sittings. University statistics have six month seasonality because of their semesters.

Jacob: Should we use the examples in the textbook for the student project?

Rachel: The last five modules in the course cover Chapter 19; these are good examples, but they are more complex, since the authors fit higher order ARIMA models.

Jacob: Should we review the earlier chapters as well?

Rachel: The entire course focuses on constructing and testing ARIMA models. The earlier chapters explain how to charts, plots, correlograms, sample and partial autocorrelation functions, regression analysis, first and second differences, and the various statistical tests to construct and choose a good model.]

Jacob: Where do we start? The textbook and postings discuss many items; is there a specific order?

Rachel: The order depends on the goals of your project and the results of previous steps. We suggest some common steps, though you vary them for your project.

Jacob: How would one start?

Rachel: Start by graphing the data. The type of graph depends on the question. To examine if monthly interest rates have a long-term drift, use a 12 month moving average. This eliminates seasonality, smooths random fluctuations, and highlights long-term trends. To examine if the stochasticity changes over time, we don’t use a moving average.

Jacob: The graph is just visual; shouldn’t we use statistical tests?

Rachel: We start with graphs. To test for random walks, we graph the interest rates and their first differences. We graph the sample autocorrelations (form a correlogram) which shows the pattern. For seasonality, we graph the interest rates and the correlogram.

Jacob: After forming the graphs and charts, do we construct an ARIMA model?

Rachel: We first see if we should separate the time series into intervals. We focus on means, drifts, and variances in each period.

Jacob: How do we know how to choose the periods?

Rachel: Choose them initially by examining the graphs, and then calculate the mean, drift, and variance of each period.

Jacob: Do we examine if the differences in the means, drifts, and variances are statistically significant?

Rachel: One might do this. The tests for statistically significant differences are not on the time series syllabus, and they are not required for the student project.

Jacob: If the means and drifts differ, must we use separate ARIMA models?

Rachel: Not necessarily. The means and drifts may differ for the original series, but not for the first or second differences. In some cases the drift may differ by time period, but the ARIMA model may help explain (and predict) the changing drift.

Jacob: What else should we examine in each project?

Rachel: We check for seasonality. We examine the correlogram, and if we observe spikes at 12 month periods, we use a seasonal ARIMA parameter and compare the models with and without the seasonal parameter. If we don’t observe spikes, we don’t need the statistical tests.

Jacob: Do we always use the Durbin-Watson statistic, Bartlett’s test, and the Box-Pierce Q statistic?

Rachel: If the student project selects one ARIMA model from a choice of two or more models, we examine these tests. In almost all student projects, at least one or two of these tests are used.

[NEAS: Graph the time series and check for drifts and seasonality. For drifts, compute moving averages. Use 12 month moving averages for monthly rates. A drift appears as a consistent upward or downward trend. A fluctuating trend may be stochasticity, not trend. For seasonality, the correlogram is a better graph. Sales volumes show large year-end changes; we need a correlogram to see slight seasonality in interest rates.

Compute deviations and see if the mean or variance is changing. If the means, drifts, or variances differ by time period, we may need to separate the time series into eras to construct ARIMA models. If the deviations are below zero on one side of the graph and above zero on the other side, the mean is changing. If the average size of the deviations changes, the variance is changing.

Several types of models are common; they have distinct graphs and correlograms. White noise shows no pattern. For white noise, examine the graph of the deviations. The devations should appear as random fluctuations about zero. The average deviation size may vary over time, suggesting a change in the variance or conditional heteroscedasticity. White noise may have a positive mean, but no trends or cycles.

A random walk takes several forms. A random walk with no drift means that an upward movement in one month is equally likely to be followed by another upward movement as by a downward movement. A trend (drift) does not negate a random walk, but we may have to detrend the rates or take first differences.]

[NEAS: The sample autocorrelation function and correlogram are powerful tools. Weak seasonality may be obscured in a scatterplot of the interest rates but may show up in the correlogram. Autoregressive, moving average, and ARMA models have distinct sample autocorrelation functions. Check for geometric decay (autoregressive), spikes (moving average), or both.]

[NEAS: Start with a 12 month difference, even if you see a six month autocorrelation.]

[NEAS: On the write-up, be sure to state the hypothesis and what the test shows.]