Step 17: Time Series Regression

What you will learn

• what makes time-series data different from a cross-section;
• how to build lags, differences and growth rates, and why log differences are growth rates;
• how to estimate a distributed lag model and interpret the impact multiplier and long-run propensity, with a standard error;
• the assumptions TS.1–TS.6, strict exogeneity, and why feedback breaks it;
• how trends create spurious regressions, and how a time trend works (the Frisch–Waugh view);
• how to model seasonality with dummies, deseasonalise a series and spot \( R^2 \) inflation;
• how to estimate and use an AR(1) model.

What makes time series different

Chapter 17 of Econometrics Made Simple covers this in full: lags, distributed lag models, trends, seasonality and the AR(1), on real US data.Get the textbook · $15

Three features separate a time series from the cross-sections of earlier steps. Ordering: in a cross-section you could shuffle the rows and nothing would change; in a time series the order is information. Dependence: if unemployment is 9% this quarter it will not be 4% next quarter, so each observation is not a fresh draw but one value on the single path history produced. That is why the errors of time-series regressions are so often autocorrelated (Step 12). Frequency and calendar effects: monthly and quarterly data keep patterns such as the December jump in retail spending, which must be modelled or removed. More frequent data also give more observations, but not necessarily more information about slow-moving relationships: 156 months is still only 13 years.

The book uses three datasets here: real quarterly US macroeconomic data, 1959Q1–2009Q3 (us_macro.csv, 203 quarters), and two simulated files, annual consumption data and monthly retail sales for 2012–2024 (156 months).

Lags, differences and growth rates

The first lag is \( y_{t-1} \), the value one period earlier; the first difference is \( \Delta y_t = y_t-y_{t-1} \); the growth rate is \( (y_t-y_{t-1})/y_{t-1} \), usually times 100. Creating a lag shifts the column down one row, so a model with \( q \) lags loses its first \( q \) observations. In pandas the tools are shift(1), diff() and pct_change(); in Stata, after tsset, the operators L., D. and F..

The approximation comes from \( \log(1+g) \approx g \) for small \( g \). Log differences also add up over time, and treat rises and falls symmetrically. For the log scale more generally, see our post on interpreting log coefficients.

Static and finite distributed lag models

A static model, \( y_t = \beta_0 + \beta_1 z_t + u_t \), assumes a change in \( z \) has its full effect at once and none afterwards. Households, though, may take several quarters to adjust their spending to a change in income.

Two thought experiments make the coefficients concrete. If \( z \) rises by one unit for one period only, \( y \) rises by \( \delta_0 \), then \( \delta_1 \), and so on, back to normal after \( q \) periods: the sequence is the lag distribution. If \( z \) rises and stays higher, the effects accumulate: \( \delta_0 \), then \( \delta_0 + \delta_1 \), and from period \( q \) onwards the LRP.

The book estimates an FDL with four lags of quarterly income growth on the real US data, where \( gc_t \) and \( gy_t \) are 100 times the log differences of real consumption and real disposable income (198 quarters):

The impact multiplier is 0.343: in the quarter that income growth is a point higher, consumption growth is about a third of a point higher. But the first lag adds 0.237 (\( t = 5.02 \)) and the third 0.098 (\( t = 2.07 \)). The LRP is \( 0.343 + 0.237 + 0.068 + 0.098-0.002 = 0.744 \), more than twice the impact effect. The static model on the same quarters has a slope of 0.342 and \( R^2 = 0.194 \): it captures less than half of the total response.

Inference on the long-run propensity

Individual lag coefficients are often imprecise because \( z_t, z_{t-1}, z_{t-2}, \dots \) are highly correlated with each other (multicollinearity, Step 10). The LRP is usually better determined, but its standard error is not the sum of the individual ones:

Software does this for you: t_test in Python, lincom in Stata, or the quadratic form with vcov() in R. Here the standard error is 0.099, so \( t = 7.51 \) and the 95% interval runs from 0.549 to 0.940. The data reject both “no long-run effect” and “complete pass-through” (LRP = 1).

Assumptions for time-series regression

Let \( \mathbf{X} \) stand for the explanatory variables in all periods, past, present and future.

The results mirror the cross-section case. Under TS.1–TS.3 OLS is unbiased; adding TS.4–TS.5 it is BLUE; adding TS.6, the usual t and F tests are exact. TS.3 is far stronger than it looks. It is not enough that \( u_t \) is unrelated to \( x_t \) (contemporaneous exogeneity). Feedback from \( y \) to future \( x \) breaks it: a city with a crime surprise this year hires more police next year; a firm whose sales disappoint raises its advertising budget. A simple check is to add the lead \( z_{t+1} \): under strict exogeneity it should not help explain \( y_t \).

In the consumption model the lead of income growth has a coefficient of 0.201 with \( t = 4.44 \). Either households anticipate income changes, as the permanent-income hypothesis predicts, or spending feeds back into income through the economy. Either way income growth is not strictly exogenous, and unbiasedness does not apply. All is not lost: under contemporaneous exogeneity and weak dependence, OLS is still consistent, which Step 18 explains.

Drill growth rates, long-run propensities and seasonal dummies with graded questions. Over 1,150 practice questions, every one with a worked answer.Practice book · $9

Trends and spurious regression

Two series that both trend upwards will be strongly correlated in levels even if neither has anything to do with the other, simply because both are high late in the sample. Leaving the trend out is an omitted-variable problem: time affects \( y \) and is correlated with \( x \). A linear trend is \( y_t = \alpha_0 + \alpha_1 t + e_t \); an exponential trend, \( \log y_t = \alpha_0 + \alpha_1 t + e_t \), suits a series growing at a steady percentage rate, with \( 100\alpha_1 \) roughly the growth rate per period. For US real GDP the exponential trend has slope 0.00790 and \( R^2 = 0.994 \): about 0.79% a quarter.

The book’s simulated retailer shows how badly a trend can mislead. Advertising was built to have no effect on sales, but both series grow and the firm advertises more in its busy months. On its own, each extra $1 million of advertising is associated with $9.67 million more sales (standard error 0.529, \( R^2 = 0.684 \)). Adding a linear trend cuts this to 2.869.

That gap is a warning about \( R^2 \): when \( y \) trends, most of the fit is the trend explaining the trend. Our post on R-squared vs adjusted R-squared explains why a high \( R^2 \) proves little on its own.

Seasonality and event dummies

Sales jump every December and dip in January and February. With monthly data and an intercept, include dummies for 11 of the 12 months; the omitted month (here January) is the base. Including all 12 is the dummy-variable trap of Step 9. The spring 2020 shop closures get an event dummy equal to 1 for March, April and May 2020.

The December effect dominates: \( \hat\gamma_{12} = 28.398 \) (standard error 1.175). At the same point on the trend, December sales are about $28.4 million higher than January’s, and \( 28.398-15.804 = 12.594 \) million higher than November’s. February is the weakest month (\( -3.013 \)). The trend is 0.444 a month, about $5.3 million a year, and sales in March–May 2020 were $20.355 million a month below normal.

Once trend and seasonality are both controlled for, the advertising effect is gone. The impressive 9.671 was built from a common trend and a common seasonal cycle. Model (4) has \( R^2 = 0.983 \), yet when both series are detrended and deseasonalised, advertising explains just 0.1% of what remains. This is \( R^2 \) inflation. The residuals of model (4) are also still positively autocorrelated (about 0.47), so robust standard errors are needed before any formal test; the Durbin–Watson test is the classic first check.

A first dynamic model: AR(1)

For quarterly US GDP growth, \( \hat\beta_1 = 0.302 \) (standard error 0.067) and \( \hat\beta_0 = 0.533 \). The implied long-run mean is \( 0.533/(1-0.302) = 0.763\% \), close to the sample mean of 0.776%. Lagged growth explains only 9.3% of the variation in growth: most quarterly movements are unpredictable. Unemployment is very different: its AR(1) coefficient is 0.988 (standard error 0.017), extremely close to 1. A series with \( \beta_1 = 1 \) never returns to a fixed mean; it has a unit root, the subject of the next step.

Because \( y_{t-1} \) is a regressor, the AR(1) cannot satisfy strict exogeneity, and \( \hat\beta_1 \) is biased in small samples, typically downwards. But if \( |\beta_1| < 1 \) and the errors are not autocorrelated, OLS is consistent, and with 201 quarters the bias is small.

Practise: 6 questions

Try each one on paper first, then check the worked answer.

Check yourself

The short version

• Time series are ordered, dependent and observed at a fixed frequency. Plot them first.
• \( 100\,\Delta\log y_t \approx \) % growth. US real GDP: 0.776% a quarter, about 3.1% a year.
• FDL: impact multiplier \( \delta_0 \), LRP \( \sum\delta_j \), whose standard error needs the covariances. US consumption: impact 0.343, LRP 0.744 (s.e. 0.099).
• TS.1–TS.3 give unbiasedness; strict exogeneity fails with feedback and always with a lagged dependent variable.
• Trends and seasons create spurious relationships and inflate \( R^2 \). Including \( t \) and season dummies is the same as detrending and deseasonalising. The fake advertising effect fell from 9.671 to 0.094.
• AR(1): long-run mean \( \beta_0/(1-\beta_1) \). GDP growth has \( \hat\beta_1 = 0.302 \); unemployment has 0.988, close to a unit root.
• Chapter 17 of Econometrics Made Simple has the full chapter with code in Python, R and Stata. If you study several subjects, the Complete Book Bundle includes every Economics, Statistics, Mathematics and Econometrics book.

Time series, unit roots and forecasting, plus every earlier chapter. Get the textbook and the practice book together for $19 (was $24).Both books · $19