# Is a time trend a substitute for first differencing?

I am doing a macroeconomic analysis involving BOP, investment ratio, GDP growth rates, and CPI inflation as dependent variables. My independent variables are other macro variables.

When I test for unit root, many of the variables display unit root.

But, by first differencing some of the IVs (for example the growth of exports) the interpretation of the affects on the DV become "growth of growth rates" and seems not right.

I know by adding a linear time trend variable as an IV into the regression, it removes the spurious regression with time. But, is this a replacement for first differencing? If not, what are the alternatives to first differencing my variables the interpretation is more coherent?

First differencing also affects the error term, so they are slightly different.

To see this, suppose you have some underlying time-series $$\{ \varepsilon_t | t \in \mathbb{Z} \}$$ and you are going to form a new time-series on top of this. Suppose we compare a series $$\{ X_t | t \in \mathbb{Z} \}$$ from a model that includes a drift term with a series $$\{ Y_t | t \in \mathbb{Z} \}$$ from a model that does not include a drift term but is defined on its first differences. Specifically, we have the two alternative models:

\begin{aligned} X_t &= \mu + \lambda t + \varepsilon_t, \\[6pt] \Delta Y_t &= \lambda + \varepsilon_t.\\[6pt] \end{aligned}

Now, taking the first difference of the first series gives:

$$\ \Delta X_t = \lambda + \varepsilon_t - \varepsilon_{t-1},$$

so the models are similar, but the error terms for the two models look different. In particular, looking at the first difference of the first model, we see that there will now be some correlation in the error terms.

• Actually, I would argue that the error terms are substantially different - $\Delta X_t$ now has zero long-run variance, i.e., is overdifferenced. Commented May 30, 2022 at 9:43

Differencing is applicable for stochastic non-stationary time series while time trends are applicable for deterministic non-stationary series. See the discussion here stochastic vs. deterministic trend in time series and ADF test fail to reject while kpss and box say white noise and stationary as to possible ways to identify which is more appropriate for any individual time series.

Decompose a time series data into deterministic trend and stochastic trend might also be of help to you .

Standard analysis of macroeconomic series is usually done via VAR (vector autoregression models), rather than fitting univariate models for individual series. Macroeconomic variables are simultaneously determined. Classification into "dependent" and "independent" are likely artificial with no economic justification.

(Structural VAR's is a slightly different story.)

VAR of integrated variables can be estimated in the same way as VAR of stationary variables, after adding more lags to accommodate the degree of integration. This circumvents the need for differencing and allows one to deal directly with levels of variables. Determining the degree of integration can be done via univariate unit root tests, as you have already carried out.

In short, if the task is standard macroeconomic analysis (e.g. forecast/impulse response/Granger causality/etc.), use VAR's. Avoid first difference unless you're specifically interested in growth rates, and avoid univariate models.