Calculate trends and seasonality using time series with regression (e.g., deterministic vs. stochastic trend).

Monthly claim counts at a homeowners book have drifted upward for ten years and spike every July. Should you model that drift as a fixed straight line, or as a random walk that just happens to have wandered up? The choice changes every forecast standard error.

Why the two trend types matter. A deterministic trend says the mean of $Y_t$ is a fixed function of time plus zero-mean noise. A stochastic trend says today's mean is yesterday's value plus a fresh shock. Both produce upward-sloping plots. Only one keeps long-horizon forecast variance bounded.

The simplest deterministic model is linear in time:

$$Y_t={\beta }_0+{\beta }_{1}t+{\epsilon }_t,{\epsilon }_t\stackrel{iid}{\sim }(0,{\sigma }^2)$$

Estimate ${\hat{\beta }}_0,{\hat{\beta }}_1$ by ordinary least squares with regressors $\{1,t\}$ for $t=1,\dots ,n$. The slope ${\hat{\beta }}_1$ is the fixed per-period drift. The $h$-step forecast is ${\hat{Y}}_{n+h}={\hat{\beta }}_0+{\hat{\beta }}_1(n+h)$ and the forecast variance approaches ${\sigma }^2$ as $n$ grows: extra time does not destabilize the level.

Common mistakes

• Including all $s$ seasonal dummies plus an intercept. This is the dummy-variable trap. The design matrix is singular and software drops a column silently. Correct count for monthly data is 11 dummies alongside the intercept.
• Fitting a linear OLS trend to a random walk and trusting the $t$-statistic. Spurious regression inflates the $t$-statistic and shrinks $p$-values toward zero. Always check residual autocorrelation; a near-unit first-order autocorrelation is the giveaway.
• Differencing a deterministic trend. First-differencing $Y_t={\beta }_0+{\beta }_{1}t+{\epsilon }_t$ gives $\nabla Y_t={\beta }_1+{\epsilon }_t-{\epsilon }_{t-1}$, a stationary MA(1) with a mean. You have thrown away the level information and induced a negative MA root.

Bottom line

• Deterministic trend: $Y_t={\beta }_0+{\beta }_{1}t+{\epsilon }_t$, fit by OLS on $t$; residuals mean-revert to the line and forecast variance stays bounded as the horizon grows.
• Stochastic trend (random walk): $Y_t=Y_{t-1}+{\epsilon }_t$ is a cumulative sum of shocks whose level never reverts, so $h$-step forecast variance grows linearly as $h{\sigma }^2$.
• Seasonal dummies: add $s-1$ indicator regressors (one season is the baseline) for additive seasonality of period $s$; each ${\hat{\gamma }}_j$ is a deviation from baseline, not a level.
• Detrend vs difference: for a deterministic (trend stationary) trend, regress on $t$ and use residuals; for a stochastic trend, first-difference $\nabla Y_t=Y_t-Y_{t-1}$.

Exam shortcut

If the problem gives time $t$ and asks for a point forecast with seasonal dummies, plug $t$ into the trend line and add the matching ${\gamma }_j$; this answers most quarterly and monthly forecast questions in one line. If the problem mentions "shocks accumulate" or "no mean reversion," default to the random walk with drift and use $h{\sigma }^2$ for forecast variance, never the flat OLS ${\sigma }^2$.

The full lesson (about 2,433 words, 16 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• D3

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