Econometrics — Questions 3 & 4
Formula Sheet
Core time-series model (AR(1))
Model, variables, and meaning
yt = α + βyt−1 + εt . (1)
• yt : value of the series at time t (e.g., inflation, GDP growth).
• α: intercept/drift; shifts the long-run mean of yt .
• β: persistence parameter; how strongly past values affect current values.
• εt : innovation (shock) at time t; assumed white noise:
– E(εt ) = 0: shocks average to zero (no systematic bias).
– Var(εt ) = σ 2 : constant shock variance.
– Cov(εt , εt−k ) = 0 for k ̸= 0: no serial correlation in shocks.
Interpretation: today’s value equals a constant, plus a fraction of yesterday’s value, plus a new
shock.
Stationarity and random walk (definitions + usage)
• Strict stationarity: the joint distribution of (yt , . . . , yt+k ) is the same as (yt+h , . . . , yt+k+h )
for all t, k, h.
• Weak stationarity: constant mean, constant variance, and covariance depends only on lag.
• For AR(1), weak stationarity requires |β| < 1.
• If |β| ≥ 1, shocks do not die out; the series is non-stationary.
• Random walk: β = 1, yt = yt−1 + α + εt .
• Random walk with drift: α ̸= 0; the mean trends over time.
Usage: exams ask whether a process is stationary and what its trend/path looks like.
MA(∞) representation (why it matters)
If |β| < 1,
∞
α
yt = + (2)
X
β j εt−j .
1 − β j=0
Meaning: current yt is a weighted sum of all past shocks, with weights β j that shrink over time.
Use: derive the mean, variance, and autocovariances directly.
1
, Moments and autocovariances (what each symbol means)
α
E(yt ) = , (3)
1−β
σ2
Var(yt ) = , (4)
1 − β2
β k σ2
Cov(yt , yt−k ) = , (5)
1 − β2
ρk = β k (autocorrelation at lag k). (6)
Here, k is the lag (how many periods apart). Use: compute autocorrelation and show weak
dependence (ρk → 0 as k → ∞ if |β| < 1).
Random walk moments (what changes)
For yt = yt−1 + α + εt with y0 given:
E(yt ) = y0 + αt, (7)
Var(yt ) = σ 2 t. (8)
Interpretation: variance grows linearly with t, so uncertainty explodes over time. Use: show
non-stationarity and compute mean/variance for a given t.
Estimation and exogeneity in time series
OLS in AR(1) (assumptions + implications)
• Strict exogeneity (needed for unbiasedness) fails in AR models because yt−1 depends on
past errors.
• Consistency requires: stationarity, weak dependence, no perfect collinearity, and contempo-
raneous exogeneity:
E(εt | yt−1 ) = 0. (9)
• If contemporaneous exogeneity fails, OLS becomes inconsistent.
• Test idea: estimate AR(1), then check residual autocorrelation (e.g., Ljung-Box); correlation
implies exogeneity failure.
Granger causality (definition + test)
In an ADL(1,1) model:
yt = α + βyt−1 + ψ0 xt + ψ1 xt−1 + ut . (10)
Test:
• H0 : ψ0 = ψ1 = 0 means x does not help predict y beyond y’s own lags.
• Rejecting H0 implies x Granger-causes y.
• Use F test for joint significance; if only one lag, F = t2 .
Use: prediction-focused causality, not structural causality.
2
Formula Sheet
Core time-series model (AR(1))
Model, variables, and meaning
yt = α + βyt−1 + εt . (1)
• yt : value of the series at time t (e.g., inflation, GDP growth).
• α: intercept/drift; shifts the long-run mean of yt .
• β: persistence parameter; how strongly past values affect current values.
• εt : innovation (shock) at time t; assumed white noise:
– E(εt ) = 0: shocks average to zero (no systematic bias).
– Var(εt ) = σ 2 : constant shock variance.
– Cov(εt , εt−k ) = 0 for k ̸= 0: no serial correlation in shocks.
Interpretation: today’s value equals a constant, plus a fraction of yesterday’s value, plus a new
shock.
Stationarity and random walk (definitions + usage)
• Strict stationarity: the joint distribution of (yt , . . . , yt+k ) is the same as (yt+h , . . . , yt+k+h )
for all t, k, h.
• Weak stationarity: constant mean, constant variance, and covariance depends only on lag.
• For AR(1), weak stationarity requires |β| < 1.
• If |β| ≥ 1, shocks do not die out; the series is non-stationary.
• Random walk: β = 1, yt = yt−1 + α + εt .
• Random walk with drift: α ̸= 0; the mean trends over time.
Usage: exams ask whether a process is stationary and what its trend/path looks like.
MA(∞) representation (why it matters)
If |β| < 1,
∞
α
yt = + (2)
X
β j εt−j .
1 − β j=0
Meaning: current yt is a weighted sum of all past shocks, with weights β j that shrink over time.
Use: derive the mean, variance, and autocovariances directly.
1
, Moments and autocovariances (what each symbol means)
α
E(yt ) = , (3)
1−β
σ2
Var(yt ) = , (4)
1 − β2
β k σ2
Cov(yt , yt−k ) = , (5)
1 − β2
ρk = β k (autocorrelation at lag k). (6)
Here, k is the lag (how many periods apart). Use: compute autocorrelation and show weak
dependence (ρk → 0 as k → ∞ if |β| < 1).
Random walk moments (what changes)
For yt = yt−1 + α + εt with y0 given:
E(yt ) = y0 + αt, (7)
Var(yt ) = σ 2 t. (8)
Interpretation: variance grows linearly with t, so uncertainty explodes over time. Use: show
non-stationarity and compute mean/variance for a given t.
Estimation and exogeneity in time series
OLS in AR(1) (assumptions + implications)
• Strict exogeneity (needed for unbiasedness) fails in AR models because yt−1 depends on
past errors.
• Consistency requires: stationarity, weak dependence, no perfect collinearity, and contempo-
raneous exogeneity:
E(εt | yt−1 ) = 0. (9)
• If contemporaneous exogeneity fails, OLS becomes inconsistent.
• Test idea: estimate AR(1), then check residual autocorrelation (e.g., Ljung-Box); correlation
implies exogeneity failure.
Granger causality (definition + test)
In an ADL(1,1) model:
yt = α + βyt−1 + ψ0 xt + ψ1 xt−1 + ut . (10)
Test:
• H0 : ψ0 = ψ1 = 0 means x does not help predict y beyond y’s own lags.
• Rejecting H0 implies x Granger-causes y.
• Use F test for joint significance; if only one lag, F = t2 .
Use: prediction-focused causality, not structural causality.
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