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Below are the sequential steps/question regarding my problem:

I am attempting to specify a VAR model in order to analyze impulse response functions. In plotting my first variable (Figure 1) I expected the seasonality because of the monthly frequency. However, the stability of the trend surprised me. I included the fitted line of the series regressed against a time trend to make things clearer. Am I correct in suspecting a deterministic rather than stochastic trend? Despite several shocks, the series reverts back to the trend line.

Figure 1

For a more formal approach, I utilized the Augmented Dickey-Fuller test including a trend and intercept (given the information in Figure 1). I selected an arbitrarily high maximum lag length of 60 and performed the test with three different information-criteria to determine the lag length (Akaike, Schwarz, and Hannan-Quinn). Both the SIC and HQ criteria suggested a lag length of 24, resulting in a 0.02 p-value (rejecting the null hypothesis of a unit root) and supporting the deterministic over the stochastic trend. The AIC suggested 29 lags, resulting in a p-value of 0.35. Given this discrepancy, I computed the ADF test for all lag lengths up to 29 (Figure 2). P-values are plotted on the vertical axis while the lag lengths are plotted horizontally. The blue bar represents the 10% significance level.

The graph clearly shows a majority of lag lengths result in a failure to reject the null hypothesis. How much weight should be placed on the lag lengths suggested by information criteria, given that so many other lags contradict the conclusion? Is there any reason to weight lags at a specific frequency higher than others (i.e. 12 and 24 lags for monthly data)?

Additionally, I have included the ADF-test results for the logged data. If the data require differencing, I would consider logarithmically transforming the data to represent the growth rate. But let's say I had a series that initially required a log transformation to stabilize its variance. Should the ADF test be performed on the log or levels?

Figure 2

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Ultimately, I would like to know if my initial identification of a deterministic trend is reasonable and to gain clarity regarding my thought process when performing the ADF test. I am currently conflicted because the initial plot appears deterministic, while a most lag lengths suggest a unit root process.

Side Note: The trend-structure of many economic series is continuously debated. Theoretically, the long-run employment trend should be stochastic because of structural changes in the economy and workforce over the past 40 years. But I greatly value empirical support, hence my current conundrum.

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