Module M09 Quiz: Serial Correlation

12 questions · Graduate (ECMT5001) · MCQ, calculation, short answer

How to use

Attempt each question before clicking Show Answer. Many serial correlation questions ask you to state H₀, compute a test statistic, and reach a conclusion.

Question 1

Lesson L01 · Definition

Serial correlation (autocorrelation) in time-series errors means:

Type: Multiple Choice

A) $\text{Var}(u_t \mid x_t) \neq \sigma^2$ (heteroskedasticity)
B) $\text{Cov}(u_t, u_{t-s}) \neq 0$ for some lag $s \neq 0$
C) $E(u_t \mid x_t) \neq 0$ (endogeneity)
D) Errors are not normally distributed

Show Answer

Answer: B

Serial correlation means past errors predict current errors — shocks persist over time. The most common pattern is positive AR(1): $u_t = \rho u_{t-1} + e_t$ with $\rho > 0$. This means: - A positive shock this period → likely positive error next period - The error process has "memory"

Common causes: omitted slowly-changing variables (e.g. technology, preferences), misspecified dynamics (lagged effects of regressors not modelled), or sluggish adjustment in economic processes.

Question 2

Lesson L01 · Consequences

With AR(1) serial correlation $u_t = \rho u_{t-1} + e_t$ ($0 < \rho < 1$), OLS is:

Type: Multiple Choice

A) Biased and inconsistent — OLS cannot be used
B) Unbiased and consistent (under strict exogeneity), but not BLUE; OLS standard errors are biased — typically understated with positive $\rho$
C) Still BLUE — serial correlation only affects efficiency when $\rho > 0.5$
D) Only a problem when $\rho > 0.9$

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Answer: B

Under strict exogeneity (which holds for most static time-series models), OLS is unbiased — serial correlation in $u$ doesn't mean $E(u_t \mid x_t) \neq 0$.

However: Gauss-Markov requires $\text{Var}(\mathbf{u}) = \sigma^2 I$ (spherical errors). With serial correlation, $\text{Var}(\mathbf{u}) \neq \sigma^2 I$ → OLS is not BLUE (GLS is more efficient). The OLS variance formula $\hat{\text{Var}}(\hat{\beta}) = \hat{\sigma}^2 (X'X)^{-1}$ is wrong — it typically understates the true variance with positive $\rho$, making $t$-statistics too large and inference too liberal.

Question 3

Lesson L02 · Durbin-Watson

The Durbin-Watson statistic is $DW \approx 2(1 - \hat{\rho})$ where $\hat{\rho}$ is the first-order autocorrelation of residuals. If $DW = 0.6$, what does this imply?

Type: Calculation

Show Answer

\[\hat{\rho} \approx 1 - DW/2 = 1 - 0.6/2 = 1 - 0.3 = \mathbf{0.7}\]

$\hat{\rho} = 0.7$ indicates strong positive serial correlation — each error predicts 70% of the next error. $DW = 0.6 \ll 2$ is well below the "no autocorrelation" value of 2.

DW ranges: - $DW \approx 2$: no serial correlation ($\hat{\rho} \approx 0$) ✓ - $DW \approx 0$: strong positive serial correlation ($\hat{\rho} \approx 1$) ✗ - $DW \approx 4$: strong negative serial correlation ($\hat{\rho} \approx -1$) ✗

Question 4

Lesson L02 · DW Limitations

The Durbin-Watson test is invalid when:

Type: Multiple Choice

A) The sample size exceeds 100
B) A lagged dependent variable $y_{t-1}$ appears as a regressor
C) The regressors are trending
D) The errors are heteroskedastic

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Answer: B

The DW test assumes that regressors are strictly exogenous. When $y_{t-1}$ is a regressor, it is correlated with $u_{t-1}$ by construction (since $y_{t-1}$ depends on $u_{t-1}$). This violates strict exogeneity and biases the DW test — it tends to be biased toward 2 (appearing to find no autocorrelation even when it exists).

Alternative: Use the Breusch-Godfrey (BG) LM test, which is valid with lagged dependent variables and can test for higher-order serial correlation ($p$ lags).

Question 5

Lesson L02 · Breusch-Godfrey Test

[Short Answer] Describe the Breusch-Godfrey test procedure for AR($p$) serial correlation.

Type: Short Answer

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Steps:

Estimate the original regression by OLS; obtain residuals $\hat{u}_t$
Regress $\hat{u}_t$ on the original regressors $\mathbf{x}_t$ and $p$ lagged residuals $\hat{u}_{t-1}, \hat{u}_{t-2}, \ldots, \hat{u}_{t-p}$
Compute LM statistic: $LM = (n-p) \cdot R^2_{\text{aux}} \sim \chi^2_p$ under $H_0$
Reject $H_0$: $\rho_1 = \rho_2 = \cdots = \rho_p = 0$ if $LM > \chi^2_{p,\alpha}$

Advantages over DW: - Valid when lagged $y$ appears as a regressor - Tests for any order $p$ of serial correlation - Based on standard asymptotic chi-squared distribution (no special tables needed)

Question 6

Lesson L02 · BG Test — Calculation

[Calculation] You run the BG test with $p = 2$ lags. The auxiliary regression of $\hat{u}_t$ on $x_t$, $\hat{u}_{t-1}$, $\hat{u}_{t-2}$ gives $R^2 = 0.12$ with $n = 100$ observations. Test at the 5% level. $\chi^2_{2, 0.05} = 5.99$.

Type: Calculation

Show Answer

\[LM = (n - p) \cdot R^2 = (100 - 2) \times 0.12 = 98 \times 0.12 = \mathbf{11.76}\]

Since $LM = 11.76 > \chi^2_{2, 0.05} = 5.99$, reject $H_0$ at the 5% level. There is significant evidence of serial correlation up to order 2. OLS standard errors are unreliable — use Newey-West (HAC) standard errors or GLS.

Question 7

Lesson L03 · Newey-West Standard Errors

Newey-West (HAC) standard errors are valid in the presence of serial correlation. They:

Type: Multiple Choice

A) Change the OLS coefficient estimates to remove the serial correlation bias
B) Correct the reported standard errors to be valid asymptotically under serial correlation (and heteroskedasticity), without changing $\hat{\beta}$
C) Are only valid for AR(1) processes
D) Require knowing the order of serial correlation

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Answer: B

Newey-West SEs use a heteroskedasticity and autocorrelation consistent (HAC) variance estimator that accounts for both serial correlation and heteroskedasticity in the OLS residuals. Like robust SEs for cross-sectional data, they correct the uncertainty estimate but not the point estimates themselves — OLS is still used for $\hat{\beta}$.

The "bandwidth" parameter determines how many lags of autocovariance are included in the estimator. A common rule of thumb: use $\lfloor 4(n/100)^{2/9} \rfloor$ lags. Valid under weak dependence (not with unit roots).

Question 8

Lesson L03 · GLS / Quasi-Differencing

Under AR(1) errors $u_t = \rho u_{t-1} + e_t$, the GLS transformation produces a model with uncorrelated errors. Describe the quasi-differenced model.

Type: Short Answer

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For the model $y_t = \beta_0 + \beta_1 x_t + u_t$ with $u_t = \rho u_{t-1} + e_t$:

Step 1: Multiply the lagged equation by $\rho$: $\rho y_{t-1} = \rho\beta_0 + \rho\beta_1 x_{t-1} + \rho u_{t-1}$

Step 2: Subtract from the original: $$\underbrace{y_t - \rho y_{t-1}}_{\tilde{y}_t} = \beta_0(1-\rho) + \beta_1 \underbrace{(x_t - \rho x_{t-1})}_{\tilde{x}_t} + \underbrace{u_t - \rho u_{t-1}}_{e_t}$$

The transformed model $\tilde{y}_t = \beta_0(1-\rho) + \beta_1 \tilde{x}_t + e_t$ has i.i.d. errors $e_t$ — OLS on this model is BLUE (GLS estimator).

Feasible GLS (FGLS): Since $\rho$ is unknown, estimate $\hat{\rho}$ from the OLS residuals: $\hat{\rho} = \sum_{t=2}^T \hat{u}_t \hat{u}_{t-1} / \sum_{t=2}^T \hat{u}_{t-1}^2$. Substitute $\hat{\rho}$ and apply the quasi-differencing transformation.

Question 9

Lesson L03 · Prais-Winsten vs Cochrane-Orcutt

Prais-Winsten estimation is preferred over Cochrane-Orcutt when:

Type: Multiple Choice

A) The sample size is large
B) Efficiency matters for small $n$ — Prais-Winsten retains the first observation (in transformed form) while Cochrane-Orcutt discards it
C) $\rho$ is close to 1 (near unit root)
D) There are multiple lags of serial correlation

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Answer: B

Cochrane-Orcutt drops the first observation after quasi-differencing (the transformed $\tilde{y}_1$ requires $y_0$ which is unavailable). Prais-Winsten keeps it using the transformation: $\tilde{y}_1 = \sqrt{1-\rho^2} \cdot y_1$, which is valid under stationarity.

For large $n$, losing one observation is negligible. For small $n$, Prais-Winsten is more efficient. When $\rho$ is close to 1, both methods struggle — the transformation becomes near-differencing and loses information.

Question 10

Lesson L01 · Detecting Serial Correlation

A residual plot of $\hat{u}_t$ against time shows long runs of positive residuals followed by long runs of negative residuals. This pattern suggests:

Type: Multiple Choice

A) Heteroskedasticity
B) Positive serial correlation — the sign of errors persists over time
C) Omitted variable bias
D) Negative serial correlation

Show Answer

Answer: B — Positive serial correlation

Long "runs" (many consecutive residuals of the same sign) are the visual signature of positive autocorrelation — each error tends to be followed by an error of the same sign. This is consistent with $\hat{\rho} > 0$ and $DW < 2$.

Negative serial correlation would appear as alternating positive/negative residuals (fast oscillation), corresponding to $\hat{\rho} < 0$ and $DW > 2$.

Heteroskedasticity appears as changing spread (fan pattern), not persistent runs of same-sign residuals.

Question 11

Lesson L03 · When to Use Each Remedy

Match each situation to the appropriate remedy for serial correlation.

Type: Short Answer

Show Answer

Situation	Recommended remedy
Serial correlation present, want valid inference with minimal assumptions	Newey-West (HAC) standard errors — valid asymptotically; simple to implement
Serial correlation present, want efficient estimates (BLUE)	FGLS / Cochrane-Orcutt / Prais-Winsten — transforms the model for efficiency
Lagged dependent variable in model, suspecting serial correlation	Breusch-Godfrey test to detect; then Newey-West SEs (GLS won't work cleanly with LDV)
Serial correlation due to stochastic trend (unit root)	First-difference the series — removes the integrated component, leaving a stationary model
Serial correlation may be due to omitted lagged variables	Add FDL terms ($x_{t-1}$, $x_{t-2}$) — correct the dynamic misspecification

Question 12

Lesson L01 · Australian Context

[Short Answer] A researcher regresses Australia's quarterly inflation on the RBA cash rate and gets significant positive serial correlation ($\hat{\rho} = 0.72$, $DW = 0.56$). What are two economic explanations for this serial correlation and what should the researcher do?

Type: Short Answer

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Economic explanations:

Omitted slowly-changing variables: Supply-side factors (oil prices, productivity growth, global inflation) change gradually and affect Australian inflation persistently but are not included as regressors. Their effects show up as correlated errors.
Misspecified dynamics / adaptive expectations: Inflation expectations are formed adaptively from past inflation, creating persistence. The static model ignores that current inflation depends on past inflation ($\pi_t = f(\pi_{t-1}, \ldots)$), leaving this persistence in the error.

What to do:

Short-run fix: Report Newey-West HAC standard errors to get valid inference without changing the model
Better model: Add lagged inflation $\pi_{t-1}$ as a regressor (or lagged cash rate) to absorb the dynamic effects — if the serial correlation disappears, the dynamic misspecification was the cause
Test formally: Run Breusch-Godfrey with $p = 4$ lags (quarterly data, seasonal consideration) before deciding on the remedy
Check stationarity: If inflation has a unit root (unusual but possible in high-inflation periods), first-differencing may be needed