Time Series

Permanent Income Hypothesis

No variable apart from current consumption should be of any value in predicting future consumption.

What is the equation for PIH?

Can we test PIH theory?

Estimate: Δc_t = α + β·Δy_{t−1} + e_t

• H₀ (PIH true): β = 0, so lagged income is insignificant, because Δc_t should reflect only news.

• If β is significant, consumption responds to predictable income (“excess sensitivity”), and PIH is rejected.

Is it true that only current consumption can explain future consumption? What if we add more lags?

If PIH is true, coefficients for further lags must not be statistically significant.

What is the null hypothesis for F-test?

H_0: y_2 = y_3 = y_4 = 0 (none of the independent variables in your model have a statistically significant relationship with the dependent variable)

When do we reject the null hypothesis for f-test?

when f-stat > critical value

What does AR stand for?

Autoregressive

What model uses past values(lags) of the forecast variable?

AR

What is the equation for AR(1) model?

$y_{t}=a_1y_{t-1}+u_{t}$ Where u_t is white noise: u_t ~ WN(0, σ²)

What does u_t ~ WN(0, σ²) mean?

E(u_t) = 0, Var(u_t) = σ², Cov(u_t, u_t+1) = 0 for all h ≠ 0

What is the condition for AR(1) to be stationary and weakly dependent?

|a_1| < 1 (the absolute value of the autoregressor must be less than 1)

What is the equation for AR(q) model?

$y_{t}=a_1y_{t-1}+a_2y_{t-2}+\cdots+a_{p}y_{t-p}+u_{t}$ where p is the autoregressive model (number of lagged terms)

What does MA stands for?

Moving average

What model uses past forecast errors in a regression-like model?

MA model

What is the equation for MA(1) model?

$y_{t}=u_{t}+\beta_1u_{t-1}$ where u_t ~ WN(0, σ²)

Is MA process always stationary and weakly dependent?

Yes

What is the equation for MA(q) model?

$y_{t}=u_{t}+\beta_1u_{t-1}+\beta_2u_{t-2}+\ldots+\beta_{q}u_{t-q}$ where q is the oder of the moving average model

How can we write MA(1) as an AR(∞)?

you use invertibility via back-substitution

How can we write AR(1) to MA(∞)?

From AR(1), we could expand the model by sequential substitution.

What is the equation for ARMA(p,q) model?

$y_{t}=a_1y_{t-2}+a_2y_{t-2}+\ldots+a_{p}y_{t-p}+u_{t}+\beta_1u_{t-1}+\beta_2u_{t-2}+\ldots+B_{q}u_{t-q}$ Where u_t ~ WN(0, σ²) and normalizer B_0 = 1.

What does ARIMA stand for?

AutoRegressive Integrated Moving Average

What is the equation for ARIMA model?

ARIMA(p,d,q) = I(d) + ARMA(p,q)

What is simulation?

Give an underlying model → simulate data series

What is analysis?

Give data series → find suitable underlying model(s).

When to use ARMA(p,q) model?

Forecast one variable using its lags (AR part) & past forecast errors (MA part). Data must be stationary and weakly dependent.

Are all AR(p) stationary?

No

When to use ARIMA(p,q) model?

Forecast one variable using its lags (AR part) & past forecast errors (MA parts). Can be used for stationary and weakly dependent series after d time differencing.

Why we look into ACFs, PACFs?

It helps us know how many terms of AR and MA parts we should include in the model, i.e., deciding what is p, q in ARIMA(p,d,q).

What is ACF?

The Autocorrelation Function (ACF) measures the linear correlation of a time series with its own past values across different time lags (h).

What is the general algebraic formula for the ACF of a stationary AR(1) model with coefficient a_1?

/$p\left(k\right)=\frac{Cov\left(y_{t,}y_{t-k}\right)}{Var\left(y_{t}\right)}=a_1^{k}$

How does the visual shape of an AR(1) ACF behave if the autoregressive parameter φ₁ is negative?

It displays an alternating (zig-zag) pattern that exponentially decays toward zero.

What is the primary difference in visual shape between the ACF of an AR(1) model and an AR(q) model?

While both tail off to zero, the AR(1) is a simple geometric decay, whereas an AR(q) can display complex damped sinusoidal waves (oscillations) and ripples.

What is the ACF of MA(1) process with coefficient B_1 for k=0?

1

What is the ACF of MA(1) process with coefficient B_1 for k=1?

$$\frac{\beta_1}{1+\beta_1^2}$$

What the ACF of MA(1) process with coefficient B_1 for k>=2?

0

What is the defining characteristic of the ACF shape for an MA(1) model?

It cuts off instantly after lag 1, meaning (p_h = 0) for all (h > 1).

At what specific lag does the ACF of an MA(q) model drop completely to zero?

It drops to zero for all lags greater than q (h > q).

If a diagnostic plot shows spikes at lags 1, 2, and 3 on the ACF, but then drops to zero, while the PACF tails off gradually, what model is suggested?

MA(3)

How is PACFs different from ACFs?

ACFs include indirect correlations, but PACFs only include direct correlations.

PACFs of AR(p) when k <= p

a_p

PACFs of AR(p) when k>p

0

What is the ACFs and PACEs of an AR(p) model?

ACFs: infinite in extent (tails off)

PACFs: go to 0 for lags larger than p

What is the ACFs and PACEs of an MA(q) model?

ACFs: go to 0 for lags larger than q

PACFs: infinite in extent (tails off)

What happens if the behavior of the ACFs and PACFs doesn’t fit any criteria for AR(p) or MA(q) model?

Use ARMA(p,q)

Is it AR(p), MA(q), or ARMA(p,q)?

AR(2)

Can MA(q) be estimated by OLS?

No PROBLEMATIC for OLS! -u_t and its lags are unobservable!

How can we estimate MA(q), ARMA(p,q) models?

Use maximum likelihood estimation (MLE).

What is the general rule when comparing AIC or BIC scores across different time series models?

The lower the score, the better the model.

Why are AIC and BIC particularly useful for identifying the order of an ARMA(p, q) model compared to looking at ACF/PACF plots?

Because for mixed ARMA models, both the ACF and PACF tail off infinitely, making visual order identification impossible. AIC/BIC provide a clear, numerical way to pick p and q.

Between AIC and BIC, which criterion places a harsher penalty on adding extra lags when the sample size is large?

BIC (because its penalty term scales with ln(n).

What happen to AIC and SBC/BIC when the fit of the model improve?

They will approach infinity.

Which model should you choose if AIC and BIC select different models?

The usual default is BIC's parsimonious choice, since BIC is consistent, should check if residuals appear to be white noise.

What does it mean if the residuals of an estimated ARMA model are successfully determined to be white noise?

It means the model has successfully extracted all predictable patterns from the data, leaving only purely random error.

In a Ljung-Box Q-test performed on ARMA residuals, do you want to reject or fail to reject the null hypothesis?

Fail to reject (you want a high p-value), because the null hypothesis states that the residuals are white noise.

What is the danger of relying on an ARMA model whose residuals are not white noise?

The standard errors will be biased, making your t-statistics, p-values, and hypothesis tests invalid.

What distributional assumption must the error terms (e_t) meet for the Durbin-Watson critical value tables to be valid?

They must be normally distributed with a mean of zero and constant variance (e_t ~ N(0, sigma^2).

What condition regarding the relationship between the regressors (X) and the errors (e) is required for the Durbin-Watson test to hold?

The regressors are strictly exogenous. E(u_t | x_1t,x_2t,…,x_kt) = 0 for all t.

What is the formal null hypothesis (H₀) of the Durbin-Watson test?

No first-order autocorrelation exists in the regression residuals (ρ = 0).

Does the Breusch-Godfrey test require strict exogeneity?

No

Question: Which of the following conditions is not required for a discrete-time stochastic process \(\{e_t\}\) to be classified as a standard white noise process?

• A) The expected value (mean) of \(e_{t}\) is equal to zero for all time periods t.

• B) The variance of \(e_{t}\) is constant and finite across all time periods.

• C) The autocovariance between \(e_{t}\) and \(e_{t-k}\) is non-zero for any lag k ≠ 0.

• D) The process satisfies the conditions for covariance stationarity.

C

Question: What does the theoretical autocorrelation function (ACF) look like for a pure white noise process at any non-zero lag k ≠ 0?

• A) It decays exponentially toward zero as the lag k increases.

• B) It is identically equal to zero for all lags k ≠ 0.

• C) It oscillates sinusoidally depending on the frequency of the series.

• D) It is equal to one for all positive lags and negative one for negative lags.

B

Question: If a time series \(Y_{t}\) is constructed as the cumulative sum of a white noise process \(e_{t}\) such that \(Y_t = Y_{t-1} + e_t\), what type of process does \(Y_{t}\) represent?

• A) A covariance stationary AR(1) process with an autoregressive coefficient less than 1.

• B) A deterministic linear trend process.

• C) A white noise process with an accumulating mean.

• D) A random walk process, which is non-stationary.

D

The model: 𝑦𝑡 = 𝛽0 + 𝛽1𝑥𝑡 + 𝑢𝑡, t = 1,2,……., n is an example of a(n):

a. autoregressive conditional heteroskedasticity model.

b. static model.

c. finite distributed lag model.

d. infinite distributed lag model.

B

3. Refer to the following model: 𝑦_𝑡 = 𝛼_0 + 𝛽_0𝑥_𝑡 + 𝛽_1𝑥_𝑡−1 + 𝛽_2𝑥_𝑡−2 + 𝛽_3𝑥_𝑡−3 + 𝑢_𝑡. In this model, 𝛽_0 + 𝛽_1 +

𝛽_2 + 𝛽_3 represents:

a. the short-run change in y given a temporary increase in 𝑥.

b. the short-run change in y given a permanent increase in 𝑥.

c. the long-run change in y given a permanent increase in 𝒙.

d. the long-run change in y given a temporary increase in 𝑥.

C

5. Suppose 𝑢_𝑡 is the error term for time period ‘t’ in a time series regression model the explanatory variables

are x_t = (x_t1, x_t2 …., x_tk). The assumption that the errors are contemporaneously homoskedastic implies that:

a. Var(𝑢𝑡|𝑥𝑡) = 𝜎²_t

b. Var(𝑢𝑡|𝑥𝑡) = ∞.

c. Var(𝑢𝑡|𝑥𝑡) = 𝜎²

d. Var(𝑢𝑡|𝑥𝑡) = 𝜎

C

6. Which of the following statements is true?

a. A model with a lagged dependent variable cannot satisfy the strict exogeneity assumption.

b. Stationarity is critical for OLS to have its standard asymptotic properties.

c. ARMA models can be estimated for nonstationary time series.

d. In an autoregressive model, the dependent variable in the current time period varies with the error term

of previous time periods.

A

7. Which of the following statements is true?

a. A random walk process is stationary.

b. The variance of a random walk process increases as a linear function of time.

c. Adding a drift term to a random walk process makes it stationary.

d. The variance of a random walk process with a drift decreases as an exponential function of time

B

8. A spurious correlation refers to a situation where:

a. two variables are related through their correlation with a third variable.

b. the correlation coefficient between two variables cannot be estimated.

c. there is direct causal relationship between two variables but tests for correlations reject this relationship.

d. the correlation between two variables is positive till the sample size reaches a threshold, and negative after the sample size crosses the threshold

A

9. The model: 𝑦𝑡 = 𝛼 + 𝜌𝑦𝑡−1 + 𝛾0𝑥𝑡 + 𝛾1𝑥𝑡−1 + 𝑣𝑡, where 𝑣𝑡 = 𝑢𝑡 − 𝜌𝑢𝑡−1 represents a:

a. finite distributed lag model.

b. Koyck distributed lag model.

c. rational distributed lag model.

d. vector error correction model.

C

10. When a series has the same average growth rate from period to period, then it can be approximated by.

a. A linear trend

b. An exponential trend

c. None of the above answers are correct

d. Both (a) and (b)

B

11. Which of the following statements about an AR(1) process is true?

a. An AR(1) process is stationary whenever the coefficient on the lagged dependent variable is greater than 1.

b. An AR(1) process is stationary when the absolute value of its autoregressive coefficient is less than 1.

c. An AR(1) process always has a finite variance regardless of the value of its autoregressive coefficient.

d. An AR(1) process cannot contain a constant term.

B

15. Consider a time series with a deterministic linear trend. Which of the following is generally an appropriate

way to remove the trend?

a. Take the first difference and assume the resulting series is always stationary.

b. Regress the series on a time trend and analyze the residuals.

c. Take the logarithm of the series; this always removes a linear trend.

d. Include the lagged dependent variable as an explanatory variable and ignore the trend.

B

What makes a time series stationary?

Its mean, variance and autocorrelation stay constant over time.

What are the main ways to transform a non-stationary series?

Differencing, detrending (regressing y_t on t and keeping the residuals), and the log transformation. They can be combined.

What problem does a log transformation fix?

Variance that grows with the level of the series (a widening “funnel”). It also turns exponential growth into a linear trend.

When can’t you use logs?

When the series has zero or negative values, such as profits.

Do logs alone remove a trend?

Usually not. You still need to difference or detrend afterwards.

What problem does detrending fix?

A deterministic trend, where the series is trend-stationary.

How do you detrend a series?

Regress y_t on t, then use the residuals: ÿ_t = y_t − α̂₀ − α̂₁t.

What does a deterministic trend look like in a plot?

The series sticks to a smooth trend line and returns to it after shocks, so shocks are temporary.

What problem does differencing fix?

A stochastic trend, or unit root, where the series is difference-stationary.

What is the first difference?

Δy_t = y_t − y_{t−1}

For a series that have a linear trend, how would you transform this series to make it stationary?

Take first difference/detrend by regressing on time variable. The trend looks linear, so no logs required.

Write the regression

Δ𝑦_𝑡 = 𝛼 + 𝜌𝑦_𝑡−1 + 𝛿_1Δ𝑦_𝑡−1 + 𝛿_2Δ𝑦_𝑡−2 + 𝛿_3Δ𝑦_𝑡−3 + 𝛿_4Δ𝑦_𝑡−4 + 𝛿_5Δ𝑦_𝑡−5 + 𝑢

What are the hypotheses of the Augmented Dicky Fuller Test?

H0: 𝜌 = 0 (the time series contains a unit root, which means the data is non-stationary)

H1: 𝜌 ≠ 0 (the time series does not contain a unit root, which means the data is stationary)

Why do we choose an ADF instead of a DF test?

We choose ADF test since the data is monthly data, so we think not just the first lag would matter, but further lags as well, so we add more lags into the test.

In what situation can we use DF test?

You’d use the plain DF test when the errors are already white noise without adding any lags.

After making the series stationary, what’s the next step?

Decide how many AR terms (p) and MA terms (q) to include, choosing between AR(p), MA(q) and ARMA(p,q).

What are the two main methods for choosing p and q?

(1) Look at the ACF and PACF. (2) Compare models using AIC and BIC.

What is the formula for the autocorrelation coefficient at lag k?

ρ(k) = Corr(y_t, y_{t−k}) = Cov(y_t, y_{t−k}) / Var(y_t)