12. Time series: serial correlation and diagnostics

01. Why can the same estimate have different uncertainty?

Monthly observations may share information; uncertainty calculations must account for it.

Return to California's 108 monthly accident counts. Suppose road conditions or enforcement stay unusual for several months. After we account for the recorded variables, the errors in adjacent months may still move together. We call correlation between errors at different dates serial correlation.

A standard error estimates how much the coefficient would move across comparable samples. Treating each month as independent can give a misleading answer if months share shocks. In this lesson, keep the accident equation in view. We will plot its residuals, check their timing and decide whether to revise the mean equation or the uncertainty calculation. The next lesson compares three standard errors for the same fit.

02. Define the monthly accident model

State the outcome, law indicators, calendar controls and error term.

We keep the monthly accident equation from Meeting 10 fixed. A is total accidents and log A its logged outcome. B and S indicate the seat-belt and speed-law states. U is unemployment and W the weekend count. Each included predictor has its own coefficient, and lowercase u is the remaining error. Keeping uppercase unemployment U distinct from lowercase error u matters in both the equation and the timing argument.

The trend term uses the monthly index t. The calendar indicators are written D subscript m t here, with gamma subscript m giving each month's shift relative to January. The sum contains eleven indicators because the intercept already represents January. We compare uncertainty procedures for this unchanged equation and sample, so a change in the standard error can be attributed to the covariance calculation rather than a new fitted comparison.

03. Follow the residual path

Look for sequences of positive or negative residuals.

Each point now shows a residual from the fitted log-accident regression. A positive residual means more log accidents than the model predicts for that month; a negative one means fewer. Reveal the sequence gradually. Do several months remain on the same side of zero? Such a run is a visual sign that nearby residuals may be related.

Residuals help us study the errors, but they are not the errors themselves. We calculate residuals after estimating the model; the population errors are unobserved. A short sequence can also show runs by chance. We will therefore use the graph to understand the concern and later use a formal test to assess it under stated assumptions.

Why would a run matter for uncertainty? Suppose one unrecorded disruption affects four consecutive months. Those observations partly repeat information about the same disruption. They need not provide as much new information as four independent disruptions. The effect on a coefficient's SE depends on which months help estimate that coefficient and how their errors are related. To see the relation more directly, we will compare each residual with the preceding month's residual.

04. See how an error can carry into the next month

An AR(1) process adds a new shock to a fraction of the previous error.

To study how errors carry over, consider \(u_t\)=\(\rho\)\(u_{t-1}\)+\(\varepsilon_t\). This is a first-order autoregressive process, abbreviated AR(1). 'Autoregressive' means that a value depends on its own past value, and 'first order' means one earlier period enters directly. Here \(u_t\) is the current error, \(\rho\), read 'rho', is the carry-over coefficient, and \(\varepsilon_t\) is a new, unpredictable change. Such a new change is called an innovation.

For this illustration, the innovations are independent and average to zero. If \(\rho\)=0.6 and last month's error is two, the part carried into this month is 1.2 before adding the new innovation. A positive \(\rho\) produces persistence: effects remain for several periods. A negative \(\rho\) tends to produce alternating signs. With |\(\rho\)|<1, meaning rho is between −1 and 1, an initial disturbance fades over time rather than growing on its own.

Move the control while keeping the same innovation draws. The graph is a simulation, data generated by a specified rule, rather than an estimated traffic-error process. It uses a burn-in: the first 300 generated periods are discarded so the starting value has little effect on the displayed path. Keeping the new shocks fixed helps us isolate what changing \(\rho\) does. The graph's vertical range adjusts to keep the path visible, so check the axis values as well as the shape.

05. Separate the comparison, model and uncertainty

Related errors do not, by themselves, tell us whether the coefficient is valid.

Read the three questions in order. First ask about the comparison. Does the policy variable move with an unobserved cause of accidents? That is a threat to exogeneity, the required restriction between the regressors and the error. Strict exogeneity asks for a mean-zero error at each date given the regressor values at all observed dates, including future ones.

Second ask about the mean model. Can earlier information still help predict accidents after we use the included variables? If so, investigate the missing part of the equation. Adding a justified lag can change the estimated comparison.

Third ask about uncertainty. Follow the arrows between disturbances in the diagram. Even when the regressors satisfy strict exogeneity, the errors can be related across months. Under the stated mean and rank conditions, OLS can still be unbiased. Its usual variance formula may miss the cross-month terms. The next slide shows these terms in a simple average before we return to regression.

06. Why dependence changes uncertainty

Related errors add covariance terms to the variance calculation.

Start with the average of n errors. To get its variance, first add the individual error variances. Then add twice the covariance for each pair of different dates. The condition s less than t counts each pair once. Finally divide by n squared, because the average divides the sum by n.

A variance measures how much one error varies around its mean. A covariance measures how two errors move together around their means. Take two errors with variance one each. With zero covariance, the variance of their average is two divided by four, or 0.5. With covariance 0.5, add twice 0.5 to the numerator. The variance becomes three divided by four, or 0.75.

A regression coefficient also combines errors, using weights determined by its regressors. The example shows why cross-date covariance matters in that weighted calculation. An unequal-variance adjustment handles different individual variances. We need another calculation to allow for cross-date terms.

07. Serial covariance enters the variance through cross-date terms

The direction of the SE change depends on both regressor and error patterns.

In simple regression with centred regressor values a t, the slope error is the sum of a t times u t divided by the sum of a t squared. Its conditional variance includes same-date error variances and cross-date covariances. The conventional independent-error formula drops those cross-date terms.

Positive error autocorrelation does not mechanically multiply every slope variance by the same number. The signs and time pattern of the regressor weights matter too. The familiar one-plus-rho over one-minus-rho multiplier describes particular persistent-mean settings under additional approximations; it is not a universal regression formula. This is why the actual traffic comparison estimates an appropriate covariance rather than imposing one generic inflation factor.

08. Check whether past information is still missing

Dynamic completeness concerns whether the model uses the relevant past information.

Suppose the residuals stay positive for several months. Ask what could have raised accidents for that long and which earlier observations might help explain it. For example, earlier travel conditions might predict current accidents even after controlling for current unemployment and the calendar. That would challenge dynamic completeness for the information we could have used.

If the timing and the economic story support a lag, fit the revised equation and compare residuals on a common sample. The first dates can drop out when a lag is added. Record that change so you can separate a new equation from a new sample.

Adding information to the mean and estimating uncertainty for the remaining dependence address different parts of the model. The residual plot gives evidence about what to investigate. Use the policy question to decide whether the revised comparison still answers it.

09. Choose the response that matches the problem

Adding lags, Newey–West and Prais–Winsten change different parts of the analysis.

Keep the mean equation and uncertainty calculation separate. Adding a lagged predictor changes what the regression uses to explain accidents; it can change coefficients, interpretation and sample size. Applying Newey–West to an unchanged equation and sample keeps the OLS coefficients while changing the estimated covariance and hence standard errors.

If a credible model describes how errors persist, another possibility is to transform the equation using that model. The next meeting derives Prais–Winsten, a transformation for an AR(1) error process, before examining its output. We do not need that method to complete today's comparison. Our task here is to decide whether remaining dependence changes uncertainty, signals missing mean dynamics, or accompanies an invalid policy comparison.

10. Durbin–Watson is a residual-difference statistic

Its familiar approximation is a descriptive guide with conditions.

The Durbin–Watson statistic, abbreviated DW, divides the sum of squared changes in fitted residuals by the sum of squared residuals. Expanding the numerator makes its connection with adjacent residual products visible. Ignoring endpoint differences gives the familiar approximation two times one minus the lag-one residual correlation.

Values below two suggest positive adjacent residual association and values above two suggest negative association. The exact testing rule has additional conditions and depends on the design; it is not a generic rule that any value different from two proves misspecification. A lagged dependent regressor complicates the classical DW setting. We therefore use the more explicit auxiliary-regression formulation of the Breusch–Godfrey test for a chosen set of residual lags.

11. A Breusch–Godfrey test keeps the original regressors in its auxiliary model

State the residual lags and the joint null.

Fit the accident equation and save its residuals. The Breusch–Godfrey test, or B G, uses a supporting regression. Predict the current residual from the original regressors and q lags of the residual. Keep the original regressors because the residuals came from that estimated model. In the displayed equation, the X term stands for the sum of those original regressor terms.

The null says all q residual-lag coefficients are zero. In the conventional L M version, multiply the supporting regression's R squared by its sample size. Under the stated reference conditions, compare the result with chi squared with q degrees of freedom.

When reporting the test, say which lag orders you used and how the software handles the initial dates. A rejection suggests that the specified residual lags still carry information. Decide whether that evidence points to a change in the mean, a dependence-aware covariance estimate, or further investigation.

12. Read the serial-correlation diagnostic

Breusch–Godfrey tests a stated no-serial-correlation null.

The Breusch–Godfrey diagnostic asks whether earlier residuals help explain the current residual after the original regressors are included. Its supporting regression adds the requested residual lags and tests their coefficients jointly. Under the null, these extra lag coefficients are zero. A rejection gives evidence against that no-serial-correlation condition for the fitted specification.

The output calls this an LM, or Lagrange multiplier, test. Its large-sample statistic uses a chi-square reference: a distribution obtained by adding squares of independent standard-normal variables. Here the degrees of freedom equal the number of residual-lag restrictions, one or four. The label lags(p) gives the lag order; this p is not a p-value. Prob greater than chi-square reports the tail probability under the null approximation. The small reported p-values support investigating dependence. They neither establish an exact AR(1) process nor diagnose policy endogeneity by themselves.

. estimates restore conventional
(results conventional are active now)

. estat bgodfrey, lags(1 4)

Breusch–Godfrey LM test for autocorrelation
---------------------------------------------------------------------------
    lags(p)  |          chi2               df                 Prob > chi2
-------------+-------------------------------------------------------------
       1     |         13.010               1                   0.0003
       4     |         16.907               4                   0.0020
---------------------------------------------------------------------------
                        H0: no serial correlation

13. Reproduce and interpret dated residual diagnostics

Try the comparison before opening the explanation.

Reproduce the diagnostics block in overleaf_technical.do using traffic.csv. Report the DW statistic and BG tests at lag orders one and four. Explain the nulls, why four months are not four quarters, and why a diagnostic result does not by itself identify a policy effect.

State the assumptions and units alongside the calculation.

14. Reproduce and interpret dated residual diagnostics: explanation

Follow the calculation and interpret its units.

Use the same 108-month regression, dates and controls. DW is about 1.200475. The BG statistics are about 13.010 for one lag (p about 0.0003) and 16.907 for four lags (p about 0.0020), rejecting their stated nulls under the reference conditions. DW describes adjacent fitted-residual differences. The BG tests jointly set the specified residual-lag coefficients to zero under their reference assumptions. Four lags mean four months. Record the actual statistics and p-values from your successful run, then distinguish a conditional-mean revision from a covariance correction and from a causal identification argument.

15. Choose an action that addresses the actual problem

Connect the observed concern to the model, comparison or uncertainty calculation.

We have seen several reasons why a regression result might need attention. Choose the next action by naming which problem is present. If the mean equation is appropriate but its errors remain related across nearby dates, a justified HAC calculation can address uncertainty. If relevant past information still predicts the outcome, reconsider whether the mean equation needs that information.

If a regressor is related to the current error, changing the SE method cannot remove that relation. For example, officials might adopt a law because they anticipate an unrecorded rise in accident risk. This is a problem with the policy comparison, even if the residual dependence is also handled carefully. The study needs a reason why its policy variation can separate the law's effect from that changing risk.

Practise linking a recommendation to the diagnosis: 'The concern is…, so I would…, assuming…'. For each case, explain what your proposed change fixes and what it leaves. This prepares you to assess a new empirical question without choosing a command just because it is labelled robust. Write one connected paragraph about the accident example: name the concern, choose a response, state its assumptions and explain what it cannot establish.

16. Separate the outcome pattern from the error pattern

Serial correlation concerns errors across dates, after specifying the model.

Further study

The raw accident graph shows the outcome before we account for its predictors. A pattern that repeats at the same part of each year is seasonality. A gradual movement across the years is a trend. The month indicators and time trend in our equation are intended to describe such patterns, so a pattern in the raw outcome is not automatically a problem in the remaining error.

We call the model's average outcome at given regressor values its conditional mean. 'Conditional' means that we specify the values of the explanatory variables when making the comparison. If the conditional-mean model is appropriate, its error averages to zero at those values. Serial correlation means that errors at different dates are correlated: for example, a positive error in one month tends to be followed by a positive error. Autocorrelation is another name for correlation across time within a series.

This distinction tells us what to examine next. First allow the model to account for trend, seasonality and the observed controls. Then inspect the residuals, the observed outcomes minus their fitted values. A seasonal accident series may have little residual dependence after this adjustment. Equally, a fairly stable accident series can leave related residuals. We need the residual graph to see what the fitted model has left unexplained.

17. Compare one month with the previous month

Put the previous month's residual on the horizontal axis.

Further study

A lag is a value from an earlier date. A lag-one residual is the previous month's residual, written eₜ₋₁. In this lag plot, the horizontal axis shows eₜ₋₁ and the vertical axis shows the current residual eₜ. Each dot therefore represents two neighboring months, rather than an outcome and a policy variable.

Consider a dot in the upper-right part of the graph. Both residuals are positive: a month with more accidents than predicted followed another such month. In the lower-left part, both are negative. A tendency for dots to occupy those two regions suggests positive lag-one association. Positive means the values tend to have the same direction relative to their averages, not that both values must always be above zero.

There are 107 pairs because the first of our 108 residuals has no observed predecessor. The graph does not estimate the law's effect. It studies the pattern left after the accident regression. We can make that pattern easier to understand with a constructed process in which we choose how strongly one error carries into the next.

18. What an SE adjustment cannot repair

HAC changes uncertainty; it does not remove a regressor's relation with the error.

Further study

The SE adjustment deals with uncertainty, but it does not change which comparison the coefficient describes. Suppose the law is introduced when officials anticipate rising accident risk that our recorded variables do not capture. The law indicator is then related to the error: the endogeneity problem defined earlier. Newey–West leaves that relation, and the OLS coefficient, unchanged.

A related problem can arise after adding last month's outcome to the equation. This is a lagged outcome. It contains the influence of last month's error. If that error is related to this month's error, the lagged outcome may also be related to the current error. OLS can then be inconsistent: collecting more months does not necessarily make it approach the target coefficient. A different SE formula cannot remove that regressor–error relation.

This gives us a practical sequence. Check the comparison and the mean model before relying on an uncertainty adjustment. If the remaining concern is error dependence under a valid coefficient model, HAC may address it. If an included variable is related to the error, we need to reconsider the model or the source of the comparison. Next we will look more closely at how omitted predictable changes can create these difficulties.

19. Apply the reasoning to quarterly inflation

Define the economic shock and time horizon before interpreting a future response.

Further study

Consider an unexpected rise in production costs, such as a disruption to oil supply. Ask how it relates to inflation one year later. Inflation is the rate at which the general price level rises. In quarterly data, a year is four observation periods, so the horizon is four.

A local projection fits a separate regression for each horizon. For horizon four, use inflation four quarters later as the outcome and today's measured cost shock as a regressor. State which information you hold fixed. Whether the slope describes a causal response depends on how the shock is measured and what else affects future inflation.

Neighbouring starting dates can share some future shocks in their errors. That is why we may use a covariance estimate that accounts for dependence. A H A C lag of twelve here reaches twelve quarters, or three years. In monthly data the same number would mean one year. Report both the frequency and the lag choice.