EMET8002 Chap.6 Time Series Data, Trends and Autocorrelation
Time Series Data, Trends and Autocorrelation
Data in time order
Week 4 turns to aggregate series observed over time, from quarterly GDP to daily stock prices.
The chapter covers time series notation, seasonality and its remedies, and the transformations that turn levels into growth rates: first differences, percentage changes and log differences.
Trends and persistence
Deterministic and stochastic trends are distinguished, and stochastic trends are shown to produce spurious regressions in which unrelated series look strongly related.
Autocorrelation is defined and computed, and HAC standard errors are introduced for autocorrelated errors.
GDP and population in the Week 6 lab
The Week 6 lab works with World Bank GDP and population data for Australia and the United States from 1960 to 2022. It rewards setting the data up with tsset, plotting each pair on a readable scale, explaining a significant levels regression as possibly spurious, and repeating the analysis in first differences and growth rates.
It then asks why detrending on a year variable is not enough, which is the step most answers miss: a stochastic trend survives it.
What this chapter covers
- 01
Time series notation and lags
- 02
Seasonality and seasonal dummies
- 03
First differences, percentage changes and log differences
- 04
Deterministic and stochastic trends
- 05
Spurious regression
- 06
Autocorrelation and persistence
- 07
HAC and Newey-West standard errors
Worked example · free
Telling a real relationship from a spurious one
- 1Both series trend upwards, so the high correlation in levels may come from shared trends rather than a link.
- 1Differencing removes a stochastic trend; the near-zero correlation of the changes shows little relationship once trends are gone.
- 1Estimate the regression in first differences or growth rates, and test for unit roots before trusting any levels result.
Key terms
- Stochastic Trend
- A random, persistent movement in a series, of which the random walk is the standard case.
- Deterministic Trend
- A trend that is a fixed function of time, removed by detrending.
- Log Difference
- The change in the natural logarithm of a series, read as an approximate growth rate.
- Autocorrelation
- The correlation of a series with its own past values.
Time Series Data, Trends and Autocorrelation FAQ
What is a spurious regression?
It is a regression in which two series appear strongly related only because both contain trends. The lecture's example is US GDP and federal debt, significant together in levels but not in first differences, and the Week 6 lab repeats the point with Australian population and US GDP.
How do you remove seasonality from monthly data?
Use seasonally adjusted data where it is published, as the ABS does for retail sales, or include dummies for eleven of the twelve months, with January as the base. The dummies measure each month's average difference from January.
Why use log differences instead of levels?
A log difference is an approximate growth rate, so it removes a stochastic trend in the level and puts series of very different size on a comparable scale. For small changes it is close to the percentage change, and it is how daily returns are defined.
What is the difference between a deterministic and a stochastic trend?
A deterministic trend is a fixed function of time, such as a straight line, and is removed by regressing the series on time and keeping the residuals. A stochastic trend is random and persistent, like a random walk, and is removed by first differences or log differences.
When are HAC standard errors needed?
When regression errors are autocorrelated, as they often are in time series because omitted factors persist over time. The lecture recommends HAC standard errors, such as the Newey-West estimator, when estimating dynamic causal effects, and notes they are usually unnecessary for pure forecasting.
Why is persistence a problem for regression?
A persistent series is correlated with its own past, and when the regression error inherits that persistence, the OLS assumption of uncorrelated errors fails. The coefficients may still be estimated, but the usual standard errors are inconsistent, so tests and confidence intervals cannot be trusted without a correction.
Assessment move
Plot every series before modelling it, then compute its first autocorrelation in levels and in differences. Practise explaining in one sentence why a levels regression of two trending series cannot be trusted. Finally, compute the first autocorrelation of one real series in levels and in growth rates, and explain the difference in a sentence that a non-specialist could follow.
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