Granger Causality

Module 9 · What did the added predictor buy us?

Gary Cornwall

Econ 6376 · The George Washington University

Part I · The predictive question

An added regressor can be informative without being a policy lever.

Where we are

$y_t=\alpha+\sum_{j=1}^{p}\phi_jy_{t-j} +\sum_{\ell=1}^{q}\beta_\ell x_{t-\ell}+\epsilon_t$

Mixing-board channel	Today
Intercept and outcome memory	Retain
Observed predictor history	Ask whether it adds information
Current $x_t$ , trend, explicit MA terms	Off in this baseline

Module 8: specify an ARDL. Module 9: compare its predictive information.

What did the added predictor buy us?

This guy again!

Clive Granger

A series $x_{1:T}$ is said to “Granger cause” $y_{1:T}$ if past values of $x_t$ have additional power in forecasting $y_t$ , after controlling for the past of $y_t$ .

How would we turn that question into a regression test?

In the ARDL for $y$ with $q$ lags of $x$ :

$H_0:\beta_1=\cdots=\beta_q=0$ , written $H_0: x\not\rightarrow y$
$H_1:$ at least one $\beta_\ell\ne0$ , written $H_1: x\rightarrow y$

Clive Granger beside a motorcycle, reproduced from the course archive.

Retains the photograph, phrase, definition, and audience question from archived F2024 lecture 9 (archive/f2024_lecture_notes/f2024_lecture_9.qmd, lines 207–212). The definition is the archive’s wording verbatim. The archive wrote the null as beta_{q,1} = … = beta_{q,s} = 0 and “H_0: x not-> y”; here it uses the course’s beta_1..beta_q from the ARDL on the “Where we are” slide. The alternative’s arrow form is added in the same style. The arrows are shorthand for the joint null on the whole included lag block, conditional on the specified information (own lags, lag lengths, any controls); they are not a claim about intervention effects. Image: archive/course_images/granger_motorcycle_photo.jpg, reused course material. The archive provides no more specific image attribution; do not invent one. Further sources: notes §§9.1–9.2 and 9.5; Hamilton §11.2, pp. 302–309. Pose the question first; the notation below it is the first step of the answer (a restriction on the x-lag block). Let students propose comparing fits with and without that block before naming the F-test. The photo has no visible caption by instructor decision; its alt text is retained.

Incremental, directional, conditional

Word	What it adds to the question
Incremental	Does $x$ help beyond the information already included?
Directional	Predicting $y$ from $x$ differs from predicting $x$ from $y$ .
Conditional	The answer depends on the other information available.

Our implementation: one-step, linear prediction with specified finite lags.

Name the outcome, horizon, and information set before naming the result.

A useful signal need not be a lever

An early report can reveal news that also changes the outcome.

The report helps a forecaster who has not observed that news.
Changing the recorded report need not change the underlying event.
Predictive evidence alone does not tell us what an intervention would do.

Would changing an initial-claims report itself change the number of unemployed people?

What happens within the period?

$y_t=\alpha+\sum_{j=1}^{p}\phi_jy_{t-j} +\underbrace{\beta_0x_t}_{\text{same period}} +\underbrace{\sum_{\ell=1}^{q}\beta_\ell x_{t-\ell}}_{\text{past }x\text{: the Granger question}}+\epsilon_t$

The Granger test keeps only $\ell\ge1$ . Granger called the $\beta_0$ channel instantaneous causality.
A past-only test can miss a real within-period effect.
Same-period dependence alone does not reveal which variable moved first.
Coarser sampling (monthly rather than weekly) pushes more responses into the $\beta_0x_t$ term.

If claims respond to a layoff wave within the same month as unemployment, which term picks it up?

The equation is a data-generating process: the working ARDL of “Where we are” with the same-period term beta_0 x_t added, so coefficients are unhatted and the disturbance is epsilon_t. The Granger null concerns beta_1..beta_q only; beta_0 is the instantaneous channel, and the past-only comparison leaves x_t out of both fits rather than testing beta_0. Sources: notes §9.3 (within-period effects and the y_t = beta_0 x_t + epsilon_t example); Granger (1969), Econometrica 37(3), 424–438, who coined “instantaneous causality” and attributed apparent instantaneous causality partly to slowness in recording information or omitted variables; Enders, 4th ed., p. 306, for a within-period shock example. Answer to the question: the beta_0 x_t (l=0) term, which the Granger test deliberately leaves out of both fits. Including x_t only in the tested block would make a joint instantaneous-plus-lagged test, a different question. A same-period correlation is symmetric, so it cannot by itself order the variables. This links to the data timing slide in Part IV.

Expectations can reverse the apparent story

Investors receive information about future dividends.

The stock price level today can incorporate that information before dividends tomorrow are observed.

Past prices help predict dividends, although prices do not affect dividends.
Past dividends do not help predict prices, although expected dividends set prices.

Which direction would the prediction test label, and what would that label mean?

Part II · The joint test

Keep the regression comparison fixed and remove one lag block.

Restricted and unrestricted fits

Unrestricted

$y_t=\hat\alpha_U+\sum_{j=1}^{p}\hat\phi_{U,j}y_{t-j} +\sum_{\ell=1}^{q}\hat\beta_{U,\ell}x_{t-\ell}+e_{U,t}$

Restricted

$y_t=\hat\alpha_R+\sum_{j=1}^{p}\hat\phi_{R,j}y_{t-j}+e_{R,t}$

Only the included $x$ -lag block is removed. Re-estimate the remaining coefficients.

The whole lag block

$H_0:\beta_1=\beta_2=\cdots=\beta_q=0$

The alternative allows at least one included cross-lag coefficient to differ from zero.

If three predictor coefficients sum to zero, have we imposed this null?

No. Offsetting coefficients can still change predictions. The null sets each included coefficient to zero.

The same comparison

Hold fixed	Why?
Outcome rows and transformations	RSS values must refer to the same target observations.
Own lags and other controls	Isolate the added predictor history.
Deterministic terms	Preserve the intended nested models.

Build the complete lagged sample before fitting either regression.

What goes wrong if the restricted fit uses four more months?

Granger causality via an F-test

$F=\frac{(RSS_R-RSS_U)/q}{RSS_U/(n-k_U)}$

Quantity	Meaning
$q$	Number of excluded predictor coefficients
$n$	Common usable observations
$k_U$	All unrestricted regression coefficients, including the intercept

Reference: $F(q,n-k_U)$ , approximately under suitable dynamic-regression conditions.

Count first, then calculate

$n=120$ . The unrestricted model has an intercept, two own lags, and three $x$ lags.

$k_U=6,\qquad q=3,\qquad n-k_U=114$

With $RSS_R=108$ and $RSS_U=90$ ,

$F=\frac{(108-90)/3}{90/(120-6)}=7.6.$

pf(7.6, df1 = 3, df2 = 114, lower.tail = FALSE)

[1] 0.0001118079

An ARDL in action

For two own lags and three predictor lags:

restricted <- lm(y ~ y_l1 + y_l2,
                 data = common, na.action = na.fail)
unrestricted <- lm(y ~ y_l1 + y_l2 + x_l1 + x_l2 + x_l3,
                   data = common, na.action = na.fail)
anova(restricted, unrestricted, test = "F")

common contains the correctly dated lag columns and the shared complete sample.

Read the fitted models before reading the p-value.

Reverse the direction with another equation

$x_t=a+\sum_{j=1}^{p_x}\rho_jx_{t-j} +\sum_{\ell=1}^{q_y}\gamma_\ell y_{t-\ell}+\epsilon_{x,t}$

Test $H_0:\gamma_1=\cdots=\gamma_{q_y}=0$ in this separate scalar regression.

Evidence can appear in one direction or both.
A nonrejection can reflect weak information, a small sample, or an unsuitable model.
Neither directional pattern identifies policy effects by itself.

An in-sample test of a predictive idea

The F-test judges a predictive idea by in-sample fit. Why not test it out of sample?

Ashley, Granger & Schmalensee (1980): rely primarily on out-of-sample forecasts.
Trade-off: in-sample fit can reward overfitting; a pre-chosen holdout costs power.

Ashley & Tsang (2014)	Cornwall et al. (2019)
Split at $\tau$ ; fit each side, predict the other; repeat over $\tau$	Hold out a block; fit on the rest, minus the next $p$ rows
Pseudo-F from prediction errors, with bootstrap inference	Posterior simulation of RMSPE/MAE; posterior odds

Both: every prediction uses data not used to fit that model, and nearly all the data are used.

Sources: Ashley and Tsang (2014), Econometrics 2(1), 72–91, doi:10.3390/econometrics2010072; Cornwall, Mills, Sauley, and Weng (2019), posted manuscript dated November 18, 2018 (intro, Section 3, conclusion); notes §9.9. Ashley, Granger, and Schmalensee (1980), Econometrica 48(5), 1149–1167, as quoted in Cornwall et al.: credible Granger-causality testing “must rely primarily on the out-of-sample forecasting performance of models relating the original (non-prewhitened) series of interest.” Ashley and Tsang’s abstract: “in-sample fit can be a poor guide to actual forecasting effectiveness.” Trade-off: Clark (2004), as cited in Cornwall et al., gives simulation evidence that out-of-sample procedures avoid spurious in-sample results from overfitting; Cornwall et al. add pre-test estimator bias. Inoue and Kilian (2005), as cited in Cornwall et al., stress the efficiency and power lost by estimating on only a sub-sample; Cornwall et al.: “This loss of power explains the continued popularity of the in-sample F-test.” Ashley and Tsang call the a priori in-sample/post-sample partition “at best, somewhat ad hoc and arbitrary.” Ashley and Tsang’s cross-sample validation (CSV): for a split tau, estimate the restricted and unrestricted models on 1:tau and predict tau+1:T, then estimate on tau+1:T and “precast” 1:tau. Summing squared prediction errors over all T periods gives RSS_R(tau) and RSS_U(tau), and a pseudo-F with the usual form. Because one tau is arbitrary, they compute it at every feasible tau and use a sample quantile; they recommend the third quartile, “CSV 75”. Their goal: “preserving most of the power of in-sample testing (by utilizing all of the sample data in the test), while also retaining most of the credibility of post-sample testing (by always basing model forecasts on data not utilized in estimating that particular model’s coefficients).” The pseudo-F is not F-distributed and can be negative when the restricted model predicts better, so inference is by bootstrap (resampling the fitting errors; a wild-bootstrap option for conditional heteroskedasticity), which requires serially independent errors. In their simulations (T = 30, 60, 120) the power cost relative to the in-sample F is “manageable”. Cornwall et al.: k-fold cross-validation, with k the held-out block size (k = 1 is leave-one-out). The held-out block and the following p rows are omitted so held-out values do not enter as lags; estimation uses observations on both sides of the block, so this is not rolling-origin forecasting. Gibbs sampling gives the exact posterior predictive distribution of RMSPE or MAE at any sample size; posterior odds compare the restricted and unrestricted models. Posterior odds are not frequentist p-values. In their simulations the test’s power is close to the in-sample F and above the Ashley–Tsang statistic. The Phillips-curve result returns in Part V. No new test is required in Core: ask which comparison matches the predictive definition on the Granger slide.

Part III · Generate the evidence

We know the data-generating mechanism. The test sees one sample.

Generating some data

$x_t=0.55x_{t-1}+\epsilon_{x,t}$ $y_t=0.4y_{t-1}+0.6x_{t-1}+\epsilon_t$

for (t in 2:total) {
  x[t] <- 0.55 * x[t - 1] + epsilon_x[t]
  y[t] <- 0.4 * y[t - 1] + 0.6 * x[t - 1] + epsilon[t]
}

Independent standard-normal innovations. Retain 600 observations after burn-in.

Which history should help predict which outcome?

Comparing the mechanism and the sample

Added history	F	p-value
$x$ in the $y$ equation	259.355	$<10^{-6}$
$y$ in the $x$ equation	1.064	0.303

The true cross-lag coefficient is 0.6. Its estimate rounds to 0.600.

Change 0.6 to zero, keep the innovations, and re-estimate. Then shorten the sample.

Fitted residuals over the 600 observations from the known one-way simulation.

An early signal of the same news

Let $z_t$ be news and $x_t$ a noisy report of it:

$x_t=z_t+\eta_t,\qquad y_t=0.4y_{t-1}+0.8z_{t-1}+\epsilon_t.$

The three disturbances are mutually independent standard-normal draws over time.

There is no $x$ term in the structural equation for $y$ .

Can yesterday’s report still improve today’s prediction?

What changes when we observe the news?

Information already available	Add $x_{t-1}$ ?
Outcome history only	The report reveals otherwise hidden news.
Outcome history and $z_{t-1}$	The news itself is already known.

What happens to the population predictive contribution of

x

It disappears in the second comparison. Its conditional population coefficient is zero. The two regressions answer different questions.

A true null can still be rejected

Information already available	Sample p-value	Population restriction
Own history only	$<10^{-6}$	$x$ adds predictive information.
Own history plus news	0.0157	$x$ adds no further information.

The second test rejects at 5%. It is a false rejection in this simulated sample.

The first finding is genuine predictive information, even though changing the recorded signal does not change the outcome mechanism.

Part IV · Claims and unemployment

Use the Module 8 data, then report the uncertainty the model leaves behind.

What exactly are we predicting?

$y_t=\Delta U_t,\qquad x_t=100\Delta\log C_t$

Object	Definition
$U_t$	Monthly unemployment rate, in percent
$C_t$	Mean of weekly initial claims whose week-ending Saturdays fall in month $t$
$y_t$	Change in unemployment, in percentage points
$x_t$	Approximately the percentage change in that monthly claims mean

Why would claims contain information beyond unemployment’s own history?

Dates on the data and dates on the clock

Training: January 1990–December 2017, 336 usable observations.
Evaluation: January 2018–December 2019, 24 observations.
Prediction begins after both previous-month measures have been released, before the target month’s unemployment release.

This is usually a nowcast of the current calendar month.

Revised historical caches support a pseudo-out-of-sample exercise, not a real-time vintage evaluation.

Timing can create or hide predictive direction

Data feature	In this exercise	Why it matters
Frequency	$C_t$ averages weekly claims before and after the mid-month CPS reference week	A monthly “lag” is not a clean calendar lead
Seasonal adjustment	BLS and DOL adjust each series separately	Filtering can add or remove lagged correlation
Release timing	Claims for month $t$ ’s reference week are public about two weeks before $U_t$	Our past-only design leaves them out on purpose

The forecast comparison therefore understates claims’ practical nowcasting value. Using $x_t$ is the same-period question from Part I.

Sources: notes §9.8; BLS CPS and DOL weekly claims release practices. The CPS unemployment rate refers to the reference week containing the 12th and is released early the following month. Weekly claims are released each Thursday for the week ending the previous Saturday, so by U_t’s release, claims covering month t’s reference week have been public for about two weeks. Aggregation and seasonal adjustment can create or remove Granger causality: Granger (1969) on slowness in recording information; Sims (1974), JASA, on seasonal adjustment; Wallis (1974), JASA, on seasonal adjustment; Marcellino (1999), JBES, on temporal aggregation; Breitung and Swanson (2002), JTSA 23(6), on temporal aggregation and causality. This is not a reason to add x_t to today’s test; it is a reason to state the information set and data construction when reporting the result.

A quick look at the data

Training data from 1990 through 2017: changes in unemployment and log changes in the monthly mean of weekly claims, on separate scales.

Working in changes is a specification choice. The stationarity checks and their qualifications are in the notes.

Fix the comparison before reading the result

Six own lags and three predictor lags in each outcome equation.

For unemployment change:

$H_0:\beta_1=\beta_2=\beta_3=0$

Count	Value
Common observations $n$	336
Restrictions $q$	3
Unrestricted coefficients $k_U$	10
Residual degrees of freedom	326

The main specification is inherited from Module 8’s exploratory work; it is not a fresh prespecified test.

The result and the diagnostic belong together

Added history and outcome	F	Nominal p	BG(12) p
Claims growth in unemployment change	21.058	$1.71\times10^{-12}$	0.0031
Unemployment change in claims growth	1.804	0.1463	0.2549

BG(12) checks residual serial correlation through twelve lags.

The reverse result depends on lag choice. Nominal p-values with $q=1$ : 0.00518 at $(p,q)=(3,1)$ and 0.0286 at $(6,1)$ ; at the main $(6,3)$ , 0.146. Fix lag lengths in advance.

How much weight can the tiny forward p-value carry when residual dynamics remain?

The forward specification fails this diagnostic. Report the nominal calculation and limitation; a validated Granger-causality verdict is not established here.

Source: notes §9.8; numerical results are the approved fixed-sample output. Third and final deliberate reveal. BG’s null is no residual serial correlation through the specified order, under its maintained assumptions. Passing a diagnostic would not prove all assumptions, and changing standard errors does not repair an inadequate conditional mean. The reverse nonrejection at (6,3) is not proof of absence, and it is lag-dependent: the notes’ fixed sensitivity menu rejects in the reverse direction with q=1 at both p=3 and p=6. Do not teach “unemployment does not predict claims.” The forward direction rejects in every row, but its shorter specifications fail BG badly (p below 1e-6), which supports stopping here rather than searching. Hamilton §11.2, p. 305: results “can be surprisingly sensitive to the choice of lag length (p) or the methods used to deal with potential nonstationarity.” Enders pp. 303 and 305: choose lag lengths in advance, e.g., by information criteria on a common sample. Values come from data/sensitivity_results.csv, generated by make_sensitivity_csv.R from the notes’ application-sensitivity chunk. Ask students to formulate a defensible sentence before revealing the interpretation. Avoid silently adding lags until both tests look favorable.

Forecast usefulness is a separate comparison

Observed unemployment and three one-step prediction series for 2018 and 2019, with fixed training coefficients.

2018–2019 RMSE

Model	RMSE
Own history	0.1129
Add claims	0.1057
No change	0.1099

Unemployment percentage points.

A modest improvement in this 24-month evaluation.

A lower in-sample RSS is mechanical. A lower held-out loss is an empirical result.

What would you report?

The transformed outcome, units, dates, horizon, and available information.
The complete lag block excluded, common sample, F statistic, and nominal p-value.
The specification choices and diagnostics that support or limit interpretation.
The separate held-out comparison, if one was made.

Rewrite “Claims cause unemployment, so reducing claims would reduce unemployment.”

Part V · Uses and limits

The predictive question travels across applications. Its interpretation still depends on the setting.

Economics and beyond

Setting	Predictive question	Interpretation issue
Money and income	Does monetary history add information about activity?	Same variables, different answer: yes for 1960s–70s data, no for 1970s–80s.
Unemployment and prices	Does labor-market history improve price prediction?	A predictive result alone does not identify a policy trade-off.
fMRI signals	Does one region’s measured history help predict another’s?	Blood-oxygen delays and sampling can affect the measured direction.

Granger causality has uses beyond economics. It retains the same information-based question.

Sources: notes §9.4; Enders, 4th ed., Ch. 5 §8, p. 307; Friedman and Kuttner (1992), AER 82(3), 472–492; Wen, Rangarajan, and Ding (2013), “Is Granger Causality a Viable Technique for Analyzing fMRI Data?”, PLoS ONE 8(7), e67428, https://doi.org/10.1371/journal.pone.0067428. The idea originates with Granger (1969), Econometrica 37(3), 424–438, and was popularized by Sims (1972), AER 62(4), 540–552, on money and income. Friedman and Kuttner hold the conditioning set fixed (nominal income, money, government spending): the money base Granger-causes nominal income in 1960Q2–1979Q2 (F = 3.68, 1% level) but not in 1970Q3–1990Q4 (F = 0.82), and the pattern is robust across monetary measures (base, M1, M2, short rates). What changes is the sample period, not the conditioning variables. Hamilton p. 305 adds that money–income results are sensitive to lag length and to the treatment of nonstationarity. Do not assert a measured decline in economics use without bibliometric evidence. fMRI measures blood-oxygen responses, not neuronal events directly; hemodynamic delays and sampling complicate interpretation, not a universal claim of invalidity.

A research connection

Cornwall, Mills, Sauley, and Weng (2019)

Predictive Testing for Granger Causality via Posterior Simulation and Cross-Validation

The Phillips-curve application finds some in-sample rejections alongside weak held-out predictive evidence.

What was predicted, against which benchmark, on which observations, and under which loss?

The method is on the Part II slide “An in-sample test of a predictive idea.” Coefficient restrictions and held-out performance measure different evidence.

Source: notes §9.9; Cornwall, Mills, Sauley, and Weng (2019), Advances in Econometrics 40A, 275–292, https://doi.org/10.1108/S0731-90532019000040A012. Posted manuscript: https://grycrnwll.github.io/files/granger.pdf, dated November 18, 2018. This slide is the empirical callback; the cross-validation method and its relation to Ashley and Tsang (2014) are on the Part II slide. Quarterly U.S. data, 1963Q1–2018Q1 (manuscript Table 6, full sample): the in-sample F rejects unemployment not Granger-causing CPI (F = 7.27) and PCE (F = 4.15) inflation measures, but not core PCE (F = 1.93, p = 0.127); the posterior odds for all three are about 1 (1.00–1.16), “much less than 4:1”, so “little evidence”. The GDP and unemployment benchmark pair is strong in both. Conclusion: “Contrary to the findings from in-sample Granger causality testing, out-of-sample testing finds little evidence of a relationship.” It studies differenced price measures; do not relabel them automatically as annualized log inflation. The Core lesson is to examine target, benchmark, information, observations, and loss, not to declare a structural Phillips-curve mechanism absent.

Key takeaways

Granger causality concerns added predictive information relative to a stated history.

Test the whole included predictor-lag block, on the same observations, using a suitable model.

A test decision can be wrong. A nonrejection is not proof of no relationship.

Diagnostics and held-out prediction matter, and neither identifies an intervention effect by itself.

Call these tests of whether

x

helps forecast

y

, not whether

x

causes

y

. — after Hamilton (1994)

Thesis line, Hamilton (1994), Time Series Analysis, §11.2, p. 307, after the oil price and recession example (pp. 306–307): “it seems best to describe these as tests of whether y helps forecast x rather than tests of whether y causes x. The tests may have implications for the latter question, but only in conjunction with other assumptions.” Hamilton’s own roles are reversed (y helps forecast x); the slide uses the course’s roles. His oil example shows the positive side: combined with the assumption that oil shocks are external, the finding that very few series Granger-cause oil prices supports a causal reading. Sources: notes Core §§9.1–9.10. Ask students to explain one of these statements with the common-driver experiment or the claims exercise. If time is short, keep the reporting discussion and these distinctions rather than adding another test or formal derivation.

Next time: cointegration and error correction

Routine lag-exclusion inference can fail for integrated levels.

Differencing everything can also discard information about a shared long-run relationship.

Module 10: cointegration and error correction, through the same scalar ARDL.

What if two series wander, but their separation stays stable?

Preview only. The notes’ optional “Looking Ahead” section supplies the algebra.

Sources: notes §9.10 and optional Looking Ahead. Enders, 4th ed., Ch. 6, pp. 358–359: the Granger representation theorem, and how Granger causality must be reinterpreted in a cointegrated system (y fails to Granger-cause z only if lagged changes in y are absent and z does not respond to the equilibrium deviation); the error-correction motivation itself is pp. 353–357. Inference readings: Enders pp. 307–309 and pp. 383–384 (with nonstationary or cointegrated variables, the Granger test is not a standard F-test). Hamilton §18.2, p. 554. Recognizing the stationarity/inference concern is Core; ECM derivation and specialized integrated-model procedures are not required here. Use Harry and Mouse’s leash as the oral preview. Algebraic rearrangement does not establish cointegration. Module 10 will examine whether a stationary combination exists and how departures from it can predict changes.