Time-Series Analysis
READING 2
TIME-SERIES ANALYSIS
EXAM FOCUS
Time series are sets of observations of a variable recorded at successive, equally spaced points in time (for example, quarterly sales revenues for a company over 60 quarters). For exam preparation, you should be able to:
- Interpret regression output for time-series models and identify violations of assumptions such as heteroskedasticity, nonstationarity, and serial correlation.
- Calculate predicted values from estimated time-series models (linear, log-linear, autoregressive, seasonal, etc.).
- Understand why a log-linear model is used for exponentially growing series, how to detect and correct seasonality, and how to interpret the root mean squared error (RMSE) for forecasting performance.
MODULE 2.1: LINEAR AND LOG-LINEAR TREND MODELS
LOS 2.a: Calculate and evaluate the predicted trend value for a time series, modeled as either a linear trend or a log-linear trend, given the estimated trend coefficients.
A time series is a sequence of observations for a variable ordered in time (for example, monthly stock returns over ten years). A series exhibits a trend if plotting the observations against time shows a consistent upward or downward movement. A pattern that repeats at regular intervals is called seasonality.
Linear Trend Model
A linear trend is represented by a straight line. An upward-sloping line indicates a positive trend; a downward-sloping line indicates a negative trend. The standard linear trend model is:
yt = b0 + b1·t + εt
Here, yt is the value of the series at time t, b0 is the intercept, b1 is the trend (slope) coefficient, and εt is the error (residual) in period t. Estimation is typically by ordinary least squares (OLS). The fitted prediction equation (with estimated coefficients) is:
ŷt = b̂0 + b̂1·t
Note that this is algebraically the same as simple linear regression; the independent variable is time, t (for example, t = 1, 2, 3, ...). For period 2, the fitted equation becomes the same formula with t = 2, and for period 3 with t = 3, etc. Each successive predicted value increases (or decreases) by the magnitude of the estimated trend coefficient, b̂1.
EXAMPLE: Using a linear trend model
Suppose you are given a linear trend model with estimated coefficients b̂0 and b̂1.
Calculate ŷt for t = 1 and t = 2.
Answer:
Using the fitted equation ŷt = b̂0 + b̂1·t, compute:
For t = 1: ŷ1 = b̂0 + b̂1·1
For t = 2: ŷ2 = b̂0 + b̂1·2
Note that the difference between ŷ2 and ŷ1 equals b̂1, the trend coefficient.
EXAMPLE: Trend analysis (manufacturing capacity utilization)
Consider hypothetical time-series data for manufacturing capacity utilization over a number of quarters. Applying OLS to fit a linear trend model to the data produces regression results. Using the estimated intercept and slope parameters, we form the prediction equation:
ŷt = b̂0 + b̂1·t
Using this equation we can generate estimated values for capacity utilization for each quarter in the sample. For example, if the estimate for the first quarter of 2020 is reported as 81.914, then:
Predicted capacity utilization for 2020.1: ŷ2020.1 = 81.914
The difference between the actual measured value (for example, actual = 82.4) and the predicted value is the residual (error) for that observation:
εt = yt - ŷt
In this case, the residual would be 82.4 - 81.914 = 0.486, a positive residual because the actual exceeds the predicted value.
Projected versus actual values are often displayed graphically: predicted values lie on the fitted straight trend line; residuals are the vertical distances between actual observations and the trend line. For example, a reported residual for t = 10 was calculated as 81.9 - 79.907 = 1.993.
Because a linear regression forces predictions to lie on a straight line, if the raw data are not linear the model may fit poorly and produce unreliable forecasts for future periods.
Log-Linear Trend Models
Many time series, particularly financial series, display exponential growth (continuous or compound growth). For a series growing at a constant percentage rate, plotting yt against t yields a convex curve. Exponential growth may be modelled as:
yt = A · (1 + g)^t · u_t (or, in continuous compounding notation, yt = A·e^{g·t}·u_t), where u_t is a multiplicative disturbance.
Taking natural logs transforms the exponential relationship into a linear one. The common log-linear specification is:
ln(yt) = b0 + b1·t + εt
Once the series is log-transformed, OLS can be applied to estimate b0 and b1. The fitted model produces a linear trend in ln(yt) and yields forecasts of ln(yt). To convert a forecast of ln(yt) back to the level yt, exponentiate the forecast (apply exp). In practice, when transforming back one may consider bias correction for the log-transformation, but for many applied contexts the unadjusted exponentiation is used as an approximate forecast.
EXAMPLE: Log-linear trend model (quarterly revenue)
An analyst estimates a log-linear trend model using quarterly revenue data (in millions of dollars) from Q1 2012 to Q4 2023 for JP Northfield, Inc. The model is:
ln(revenue_t) = b̂0 + b̂1·t
To forecast revenue for Q1 2024 (the next quarter), note that if the sample had 48 observations then t for Q1 2024 equals 49. The forecast in log-terms is:
ln(ŷ49) = b̂0 + b̂1·49
To convert to revenue:
ŷ49 = exp(ln(ŷ49))
For example, if ln(ŷ49) = 8.41, then exponentiating gives revenue ≈ exp(8.41) ≈ 4,492 million (using the calculator function exp or the [2nd] ex on BA II Plus after entering 8.41).
LOS 2.b: Factors that determine whether a linear or a log-linear trend should be used; limitations of trend models
Factors that determine which model is best
The analyst should begin by plotting the data:
- If the data plot roughly along a straight line with observations approximately equally distributed above and below the line, a linear trend may be appropriate. Examples: some inflation series or variables that increase by roughly constant absolute amounts over time.
- If the data plot in a curved (convex or concave) pattern suggesting constant percentage growth or decay, a log-linear trend is likely more suitable. Financial time-series, stock indices and many company sales series often follow log-linear trends.
- If residuals from a linear trend show persistent positive or negative patterns (serial correlation), the log transformation may reduce serial correlation and improve fit.
When a variable grows at a constant percentage rate, use a log-linear model. When it increases by roughly constant absolute amounts, use a linear trend model.
Limitations of trend models
Trend models rely on assumptions of linear regression. A key assumption is that residuals are uncorrelated. Violation of this assumption (autocorrelation) means OLS standard errors and hypothesis tests are invalid. In autoregressive (AR) contexts, serial correlation in residuals indicates model misspecification: the trend model would not be appropriate for forecasting, and an AR or other dynamic model should be considered.
The Durbin-Watson (DW) statistic is commonly used to detect first-order autocorrelation in residuals from models where the independent variable is time. A DW near 2 suggests no first-order serial correlation; values substantially below or above 2 suggest positive or negative autocorrelation respectively. However, the DW test is not appropriate for testing autocorrelation in residuals from AR models that already include lagged dependent variables; other tests (see Module 2.2) are used there.
MODULE QUIZ 2.1
Use the following information to answer Questions 1 through 4. Consider the results of the regression of monthly real estate loans (RE) in billions of dollars by commercial banks over the period January 2020 through September 2023 in the following table: Time Series Regression Results for Real Estate Loans
1.
The regression of real estate loans against time is a(n):
A.
trend model.
B.
AR model.
C.
ARCH model.
2.
The results of the estimation indicate an:
A.
upward trend.
B.
AR(2) model.
C.
ARCH system.
3.
Are the intercept and slope coefficient significantly different from zero at the 5% level of significance?
A.
Both are statistically significant.
B.
One is, but the other is not.
C.
Neither of them is statistically significant.
4.
The forecasted value of real estate loans for October 2023 is closest to:
A.
$1,733.764 billion.
B.
$1,745.990 billion.
C.
$1,758.225 billion.
5.
An analyst has determined that monthly sport utility vehicle (SUV) sales in the United States have been increasing over the last 10 years, but the growth rate over that period has been relatively constant. Which model is most appropriate to predict future SUV sales?
A. SUVsalest = b0 + b1(t) + et.
B. lnSUVsalest = b0 + b1(t) + et.
C. lnSUVsalest = b0 + b1(SUVsalest-1) + et.
MODULE 2.2: AUTOREGRESSIVE (AR) MODELS
LOS 2.c: Explain the requirement for a time series to be covariance stationary and describe the significance of a series that is not stationary.
An autoregressive (AR) model regresses the dependent variable on lagged values of itself. For example, sales at time t may be modeled using sales at time t-1. A simple AR(1) model is:
xt = b0 + b1·xt-1 + εt
Statistical inference using OLS for AR models is valid only when the series is covariance stationary. A time series is covariance stationary if:
- Constant and finite expected value: E(xt) = μ for all t (the series reverts to a fixed mean);
- Constant and finite variance: Var(xt) is finite and does not change over time;
- Constant and finite covariance at given lags: Cov(xt, xt-k) depends only on k, not on t.
LOS 2.d: Describe AR(p) structure and calculate one- and two-period-ahead forecasts
An AR(p) model includes p lagged values of the dependent variable as regressors. The general AR(p) is:
xt = b0 + b1·xt-1 + b2·xt-2 + ... + bp·xt-p + εt
Forecasting with AR models uses the chain rule of forecasting: produce a one-step-ahead forecast first, then use that forecast to produce a two-step-ahead forecast, and so on. For an AR(1) model:
One-period-ahead forecast:
ŷt+1|t = b̂0 + b̂1·xt
Two-period-ahead forecast:
ŷt+2|t = b̂0 + b̂1·ŷt+1|t
Note that the hat (^) indicates estimated or forecast values; multi-period forecasts embed previous forecasts as inputs and therefore are less certain than one-step forecasts.
EXAMPLE: Forecasting with AR(1)
Suppose an estimated AR(1) model is:
xt = 1.2 + 0.45·xt-1
If the current value (at time t) is xt = 5.0, calculate a two-step-ahead forecast (ŷt+2|t).
Solution (step-wise):
Compute the one-step-ahead forecast:
ŷt+1|t = 1.2 + 0.45·5.0
Compute the numerical value for ŷt+1|t.
Compute the two-step-ahead forecast using ŷt+1|t as input:
ŷt+2|t = 1.2 + 0.45·ŷt+1|t
Compute the numerical value for ŷt+2|t.
(Perform the arithmetic to obtain the final numeric forecasts.)
LOS 2.e: Use autocorrelations of residuals to test AR model fit
When an AR model is correctly specified, residuals should not display serial correlation. Serial correlation in residuals leads to unreliable standard errors and incorrect t-tests for coefficients. The general procedure to test specification of an AR model is:
- Estimate the AR model by linear regression.
- Compute the autocorrelations (sample autocorrelation function, ACF) of the residuals.
- Test whether the autocorrelations are significantly different from zero.
For each lag k, test H0: ρk = 0 using a t-statistic approximated as:
t = ρ̂k / SE(ρ̂k)
where the standard error of the sample autocorrelation is approximately 1/√T for large T (more precisely √(1/T) for the simple case), and T is the number of observations. If the absolute value of the t-statistic is greater than the critical t-value (e.g., 1.98 for a two-tailed test at 5% with roughly 100 degrees of freedom), then the autocorrelation at that lag is statistically significant, indicating model misspecification. Add appropriate lags to the model and retest until residual autocorrelations are insignificantly different from zero.
Professor's note: The Durbin-Watson statistic used with trend models is not appropriate for models that include lagged dependent variables (AR models). Use the t-test on residual autocorrelations instead.
EXAMPLE: Testing AR(1) residuals for serial correlation
Given residual autocorrelations from an AR(1) model estimated with T = 102 observations, suppose the sample autocorrelation at lag 2 is 0.0843368. The standard error is approximately 1/√T = 1/√102 ≈ 0.099. Compute the t-statistic:
t = 0.0843368 / 0.099 = 0.8518
The two-tailed critical t-value at 5% with 100 degrees of freedom is approximately 1.98. Since |0.8518| < 1.98,="" the="" lag-2="" autocorrelation="" is="" not="" statistically="" significant.="" if="" none="" of="" the="" residual="" autocorrelations="" are="" significant,="" there="" is="" no="" evidence="" of="" serial="" correlation="" and="" the="" ar(1)="" model="" may="" be="" correctly="">
LOS 2.f: Mean reversion and mean-reverting level
A time series is mean reverting if it tends to move toward its long-run mean: when above the mean it tends to fall; when below it tends to rise. For an AR(1) model:
xt = b0 + b1·xt-1
If the series is at its mean-reverting level μ, the model implies μ = b0 + b1·μ. Solve for μ:
μ = b0 / (1 - b1)
An AR(1) series has a finite mean-reverting level only when |b1| < 1.="" if="" b1=""> 0 and less than 1, the series will move gradually toward μ; if b1 is negative and |b1| < 1,="" the="" series="" will="" oscillate="" around="">
EXAMPLE: Mean-reverting level (manufacturing capacity utilization)
Using regression results for manufacturing capacity utilization an estimated mean-reverting level was reported as 67.16. This implies that if the current level is above 67.16 it is expected to fall in the next period; if below 67.16 it is expected to rise in the next period.
LOS 2.g: In-sample vs out-of-sample forecasts and RMSE for comparing models
In-sample forecasts are predictions for observations within the range of data used to estimate the model; errors are εt = yt - ŷt for t inside the sample. Comparing predicted vs actual values on the sample provides indication of fit but not necessarily predictive power.
Out-of-sample forecasts are predictions for periods beyond the estimation sample. Out-of-sample performance more directly reflects real-world forecasting ability.
Root mean squared error (RMSE) is commonly used to compare forecasting accuracy for out-of-sample predictions. For an evaluation sample of N out-of-sample observations with forecast errors e1, e2, ..., eN, RMSE is:
RMSE = √( (1/N) Σ ei^2 )
The model with the smaller out-of-sample RMSE is preferred for forecasting even if it had a higher in-sample R^2.
LOS 2.h: Instability of coefficients in time-series models
Economic and financial time series often exhibit instability or nonstationarity: underlying processes evolve over time (policy changes, structural breaks, technological changes). Coefficients estimated over long samples may be unstable if the economic environment changes. There is a trade-off:
- Longer samples provide more data and higher statistical precision.
- Shorter samples may produce more stable estimates if the underlying process is relatively constant over that shorter period.
Before trusting a model, consider whether regulatory, structural, or other changes make earlier data less relevant. Testing for covariance stationarity and for structural breaks is essential; mere absence of residual autocorrelation does not guarantee model validity.
MODULE QUIZ 2.2
1.
Is the time series shown in the following figure likely to be covariance stationary?
A.
X is not covariance stationary due to homoskedasticity.
B.
X is not covariance stationary due to non-constant mean.
C.
X is covariance stationary.
2.
Given the prediction equation: what is the forecast value of
is 16.5?
A.
64.28.
B.
117.49.
C.
210.61.
3.
When evaluating a time series model's real-world ability to forecast, we would have the most confidence in a model with small:
A.
in-sample forecast error.
B.
out-of-sample forecast error.
C.
residuals.
MODULE 2.3: RANDOM WALKS AND UNIT ROOTS
LOS 2.i: Characteristics of random walks vs covariance stationary processes
A series is a random walk if the best predictor of xt is its previous value plus an unpredictable shock. A simple random walk is:
xt = xt-1 + εt
where:
- E(εt) = 0
- Var(εt) = σ^2
- Cov(εi, εj) = 0 for i ≠ j
A random walk with a drift includes a constant term:
xt = c + xt-1 + εt
Random walks (with or without drift) are nonstationary. They have no finite mean-reverting level because solving μ = c + μ gives division by zero if considering (1 - 1) in the denominator; equivalently the AR(1) coefficient equals 1 (a unit root), so |b1| = 1 and the usual stationary conditions fail.
LOS 2.j and LOS 2.k: Unit roots - implications, testing (Dickey-Fuller), and first differencing
If the coefficient on the lagged variable in an AR(1) equals one, the series contains a unit root and behaves like a random walk. Ordinary regression inference is invalid for nonstationary series unless transformed. Two approaches to assess stationarity are:
- Estimate an AR model and examine the pattern (and significance) of residual autocorrelations.
- Apply the Dickey-Fuller (DF) test to formally test for a unit root.
Dickey and Fuller transform the AR(1) model xt = b1·xt-1 + εt by subtracting xt-1 from both sides to obtain the regression in first differences:
Δxt = (b1 - 1)·xt-1 + εt
The DF test assesses whether g ≡ (b1 - 1) = 0. The null hypothesis is g = 0 (i.e., b1 = 1, the series has a unit root). The DF test uses modified critical values (nonstandard distribution). For exam purposes, understand the test logic: if the null cannot be rejected, the series has a unit root; if the null is rejected, it does not.
First differencing
If a series is a random walk (has a unit root), applying first differencing often yields a covariance-stationary series. Define:
yt = Δxt = xt - xt-1
If xt follows a random walk xt = xt-1 + εt, then yt = εt, which is stationary (white noise). More generally, after first differencing you can model yt by an AR model, for example:
yt = b0 + b1·yt-1 + υt
Under many circumstances this transformed series is covariance stationary and suitable for OLS estimation.
EXAMPLE: Unit root and first differencing (manufacturing capacity utilization)
If capacity utilization appears to contain a unit root, transform the series by first differencing and fit an AR model to the differences. After differencing, the sample size is reduced by one (for example 40 original observations produce 39 first differences). Regress the first differences yt on their lag yt-1 to obtain an AR(1) for the differenced series. If the coefficient on yt-1 is statistically significant and residual autocorrelations are insignificant, the differenced model is acceptable.
MODULE QUIZ 2.3
Use the following information to answer Questions 1 and 2. The results of the estimation of monthly revolving credit outstanding (RCO) on the one-period lagged values for RCO from January 2020 through December 2022 are presented in the following table.
1.
What type of time-series model was used to produce the regression results in the table? A(n):
A.
AR model.
B.
heteroskedasticity (H) model.
C.
trend model with a drift.
2.
An approach that may work in the case of modeling a time series that has a unit root is to:
A.
use an ARCH model.
B.
use a trend model.
C.
model the first differences of the time series.
3.
Which of the following will always have a finite mean-reverting level?
A.
A covariance-stationary time series.
B.
A random-walk-with-drift time series.
C.
A time series with unit root.
4.
Which of the following statements is most accurate? A random walk process:
A.
is nonstationary.
B.
has a finite mean-reverting level.
C.
can be appropriately fit as an AR(1) model.
5.
Which of the following is not correct about the Dickey-Fuller unit root test for nonstationarity?
A.
The null hypothesis is that the time series has a unit root.
B.
A hypothesis test is conducted using critical values computed by Dickey and Fuller in place of conventional t-test values.
C.
If the test statistic is significant, we conclude that the times series is nonstationary.
MODULE 2.4: SEASONALITY
LOS 2.l: Test and correct for seasonality; forecast using AR model with seasonal lag
Seasonality is a recurring pattern at fixed periodic intervals (for example monthly or quarterly patterns repeating each year). For instance, monthly retail sales often peak in December. When seasonality exists, failing to include seasonal structure in the model leads to misspecification and poor forecasts.
Seasonality is often detected by inspecting residual autocorrelations at lags corresponding to the seasonal period: e.g., lag-12 for monthly data, lag-4 for quarterly data. A significantly nonzero autocorrelation at the seasonal lag indicates seasonality.
EXAMPLE: Detecting seasonality (quarterly hotel occupancy)
Suppose you model log quarterly occupancy for a resort chain with an AR(1) model using 40 quarters (which yields 39 usable differences). Examine residual autocorrelations for lags 1-4. If the lag-4 autocorrelation is large and its t-statistic exceeds the critical value (for example, with 37 degrees of freedom the two-tailed 5% critical t ≈ 2.026), then seasonality is present. For example, if the lag-4 t-statistic is 5.4460, reject H0 that the lag-4 autocorrelation is zero and conclude seasonality exists. The model must be augmented to include a seasonal term (a lag corresponding to the same quarter in the prior year).
Professor's note: With 40 points the first differences produce 39 observations: differences are between consecutive observations, so the number of differences is one fewer than the number of raw data points.
Correcting for seasonality
To correct for annual seasonality in quarterly data, include the lag corresponding to the same quarter last year. For example, the seasonally adjusted AR model for ln(xt) may be specified as:
ln(xt) = b0 + b1·ln(xt-1) + b2·ln(xt-4) + εt
Note: including ln(xt-4) does not make the model an AR(2) in the usual sense of consecutive lags; it is an AR(1) with an additional seasonal lag term. Re-estimate the model and check residual autocorrelations. If inclusion of the seasonal lag removes the significant residual autocorrelation at the seasonal lag and improves adjusted R^2 (for example from 79.3% to 94.9% in an illustrative case), the seasonal term has corrected the misspecification.
Forecasting using an AR model with a seasonal lag
Once estimated, such a model can produce forecasts. For example, suppose the seasonally adjusted fitted model in log-terms is:
ln(xt) = b̂0 + b̂1·ln(xt-1) + b̂2·ln(xt-4)
To forecast the hotel occupancy for Q1 2023, substitute ln(xt-1) (the prior quarter) and ln(xt-4) (the same quarter one year ago) into the fitted equation, compute ln(ŷt), and then exponentiate to obtain ŷt (the occupancy level). For example, a computed ln(ŷt) = 13.3103 corresponds to ŷt ≈ exp(13.3103) ≈ 603,379 (rounded), indicating a substantial increase compared with the same quarter in the prior year.
Professor's note: To convert the log forecast to a level on a BA II Plus, enter 13.3103 and press [2nd] ex to compute exp(13.3103) = 603,378.52.
MODULE QUIZ 2.4
Use the following information to answer Questions 1 through 3.
1.
The number of observations in the time series used to estimate the model represented in the table is closest to:
A.
16.
B.
50.
C.
250.
2.
Based on the information given, what type of model was used?
A.
AR(1).
B.
AR(2).
C.
AR(12).
3.
At a 5% level of significance, does the information indicate the presence of seasonality?
A.
No, because the lag-12 autocorrelation of the residual is not significant.
B.
Yes, because the lag-12 autocorrelation of the residual is significantly different than one.
C.
There is not enough information provided; the autocorrelation for the first lag is also needed to detect seasonality.
4.
A time-series model that uses quarterly data exhibits seasonality if the fourth autocorrelation of the error term:
A.
differs significantly from 0.
B.
does not differ significantly from 0.
C.
does not differ significantly from the first autocorrelation of the error term.
5.
In an autoregressive time-series model, seasonality may be corrected by:
A.
excluding one or more of the lagged variables until the seasonality disappears.
B.
transforming the time series using first-differencing.
C.
adding an additional variable that reflects an appropriate lag of the time series.
6.
Which of the following AR models is most appropriate for a time series with annual seasonality using quarterly observations?
A.
b1xt-1 + b2xt-12 + εt.
B.
b0 + b1xt-1 + b2xt-4 + εt.
C.
b0 + b1xt-4 + b2xt-12 + εt.
MODULE 2.5: ARCH AND MULTIPLE TIME SERIES
LOS 2.m: Explain autoregressive conditional heteroskedasticity (ARCH) and using ARCH to predict variance
Autoregressive conditional heteroskedasticity (ARCH) exists when the variance of residuals in one period depends on the variance observed in prior periods. When ARCH is present, OLS standard errors for mean equations are invalid. The ARCH framework models the conditional variance directly. An ARCH(1) specification regresses the squared residuals on their first lag:
εt^2 = a0 + a1·εt-1^2 + μt
If the estimated coefficient â1 is statistically different from zero, the series exhibits ARCH(1). If ARCH is present, use estimation methods robust to heteroskedasticity (for example, generalized least squares or maximum likelihood with an appropriate variance specification) to obtain valid inference for the mean equation.
Predicting the variance with ARCH models
An estimated ARCH(1) model can be used to predict the variance of residuals in the next period. If the model is:
σt^2 = a0 + a1·εt-1^2
then the one-period-ahead forecast of the conditional variance is:
σ̂t+1^2 = â0 + â1·εt^2
For example, if â0 and â1 are estimated and the current-period squared residual is reported as 0.5625, substitute into the equation to compute the predicted variance in the next period.
LOS 2.n: Nonstationarity and cointegration in multiple time-series regressions
When regressing one time series on another (for example, using the market model to estimate equity beta: yt on xt), either or both series may be nonstationary. The analyst should:
- Perform unit-root (Dickey-Fuller) tests on each series separately.
- Classify results into five scenarios:
- Both series are covariance stationary (no unit roots) - standard regression is valid.
- Only the dependent variable is stationary - regression is unreliable.
- Only the independent variable is stationary - regression is unreliable.
- Neither series is stationary and the series are not cointegrated - regression is unreliable (spurious).
- Neither series is stationary and they are cointegrated - regression residuals are stationary, and regression is meaningful.
Cointegration occurs when two or more nonstationary series have a stable long-run relationship so that their linear combination (residuals from regressing one on the other) is stationary. To test for cointegration, regress yt on xt and apply the Dickey-Fuller test to the residuals (Engle-Granger approach). If the DF test on residuals rejects the null of unit root, the residuals are stationary and the series are cointegrated; the regression is then valid despite nonstationarity of individual series.
Professor's note on DF-EG
For cointegration testing (DF-EG), critical values are nonstandard (Engle and Granger) rather than the usual t critical values. For exam purposes, you do not need to memorise these critical values; instead understand the logic: rejecting the null on the residuals implies cointegration and validity of the regression.
LOS 2.o: Choosing an appropriate time-series model
Guidelines to choose an appropriate model:
- Decide the objective: model a variable over time (trend or AR analysis) or model relationships between series (cointegration, multiple regression).
- Plot the series and inspect for nonstationarity characteristics: non-constant mean, non-constant variance (heteroskedasticity), seasonality, and structural breaks.
- If no seasonality or structural break and the plot is linear: use a linear trend model; if the plot is curved suggesting constant growth rate: use log-linear trend.
- Estimate the trend, compute residuals, and test for serial correlation (Durbin-Watson for simple trend models). If no serial correlation is detected, the trend model may be acceptable.
- If serial correlation is present, consider AR models but first check stationarity. If nonstationary, transform the data as follows:
- If linear trend: first-difference the data.
- If exponential trend: first-difference the logged data.
- If structural break: consider fitting separate models before and after the break.
- If seasonal: include an appropriate seasonal lag term.
- After transforming and differencing, if the series is covariance stationary, estimate an AR(1) model and test residuals for serial correlation and seasonality. If serial correlation persists, include additional lags (including seasonal lag) until residual autocorrelations are insignificant.
- Test for ARCH by regressing squared residuals on their lagged values. If significant, correct using generalized least squares or appropriate volatility models (ARCH/GARCH) for valid inference.
- When comparing two reliable forecasting models, compare their out-of-sample RMSE; the model with lower out-of-sample RMSE is preferred for forecasting.
MODULE QUIZ 2.5
1.
Which of the following is true of modeling a time series that contains two or more distinct periods where the data is fundamentally different?
A.
The optimal data sample period for estimating the time-series model can be calculated mathematically.
B.
To most accurately estimate the time-series model, the entire available time series data set should be used as the sample period.
C.
We have to fit two different models for each of the two distinct periods.
2.
Which of the following indicates the presence of Autoregressive Conditional Heteroskedasticity (ARCH) in a time-series model?
A.
The autocorrelations of the error terms are zero at all lags.
B.
The variance of the current error depends on the variance of lagged errors.
C.
The error term shows significant serial correlation at lag 1.
3.
Linear regression is least appropriate for modeling the relationship between two time series when:
A.
neither series has a unit root.
B.
one of the time series has a unit root, the other does not.
C.
both series have a unit root, and the time series are cointegrated.
KEY CONCEPTS
LOS 2.a
A time series is a set of observations for a variable over successive periods. A time-series model captures patterns such as trend, seasonality, and autocorrelation and allows prediction of future values.
LOS 2.b
Simple linear trend: yt = b0 + b1·t + εt, for t = 1, 2, ..., T.
Log-linear trend: ln(yt) = b0 + b1·t + εt, appropriate when data grows at a constant percentage rate.
Use a plot of the data to decide between linear and log-linear specifications. Trend models are limited when residuals show serial correlation; in that case AR or other dynamic models are preferable.
LOS 2.c
A time series is covariance stationary when its mean, variance, and covariances with lagged values are constant over time. Covariance stationarity is required for reliable inference in AR models.
LOS 2.d
Autoregressive forecasts proceed stepwise: compute one-step-ahead forecasts, then use them to compute the next-step forecasts. This is called the chain rule of forecasting.
LOS 2.e
Procedure to test whether an AR model fits: estimate the model, compute residual autocorrelations, and test whether those autocorrelations are significantly different from zero. Significant autocorrelations indicate misspecification and the need for additional lags.
LOS 2.f
A time series is mean reverting if it tends toward its mean over time. For an AR(1), the mean-reverting level is μ = b0 / (1 - b1) (requires |b1| <>
LOS 2.g
In-sample forecasts use the sample data to evaluate fit; out-of-sample forecasts evaluate predictive performance on data not used for estimation. Use RMSE on out-of-sample data to compare forecasting accuracy of competing models; the lower RMSE is preferred.
LOS 2.h
Many economic and financial time series are nonstationary. Coefficient stability depends on sample length and changes in the economic environment; shorter samples may produce more stable estimates when structural changes occur.
LOS 2.i
A random walk: xt = xt-1 + εt. Random walks do not have finite mean-reverting levels and are nonstationary.
LOS 2.j
A series has a unit root if the coefficient on the lagged dependent variable equals one. Series with unit roots are not covariance stationary. Many economic and financial series have unit roots and must be differenced.
LOS 2.k
To assess covariance stationarity: (1) estimate AR models and examine residual autocorrelations, and/or (2) perform the Dickey-Fuller test for unit roots.
LOS 2.l
Seasonality is detected by testing residual autocorrelations at seasonal lags. If significant, include the seasonal lag in the AR model. If inclusion corrects significant autocorrelations, the model is appropriately specified.
LOS 2.m
ARCH exists if residual variances are serially correlated. Test by regressing squared residuals on lagged squared residuals. If a1 is significant, ARCH exists and future variance can be predicted using σ̂t+1^2 = â0 + â1·εt^2.
LOS 2.n
When regressing two time series: if neither has a unit root, standard regression is valid; if only one has a unit root, regression is invalid; if both have unit roots but are cointegrated, regression is valid; if both have unit roots and are not cointegrated, regression is invalid. Use DF-EG test for cointegration.
LOS 2.o
Use RMSE to compare forecasting performance of competing models; choose the model with lower out-of-sample RMSE.
ANSWER KEY FOR MODULE QUIZZES
Module Quiz 2.1
1.
A With a trend model, the independent variable is time, t. (LOS 2.b)
2.
A The slope coefficient (b1) is positive and significantly different from zero indicating an upward trend. (LOS 2.a)
3.
A The t-statistic to test the statistical significance of the intercept and slope coefficient is the parameter estimate divided by its standard error. We reject the null hypothesis and conclude the coefficients are statistically significant if the absolute value of the t-statistic is greater than the two-tail 5% critical t-value with 43 degrees of freedom, which is 2.02. Both the intercept term and the slope coefficient are significantly different from zero at the 5% level because both t-statistics are greater than the critical t-value of 2.02. (LOS 2.a)
4.
C = $1,195.6241 + $12.230448(46) = $1,758.225 billion (LOS 2.a)
5.
B A log-linear model (choice B) is most appropriate for a time series that grows at a relatively constant growth rate. Neither a linear trend model (choice A), nor an AR(1) model (choice C) are appropriate in this case. (LOS 2.b)
Module Quiz 2.2
1.
B Time series X has a definite upward trend, which suggests the expected value of the time series X is not constant, and therefore it is not covariance stationary. (LOS 2.c)
2.
B
3.
B Out-of-sample performance is the most important indicator of a model's real-world forecasting ability. In-sample forecast performance is less persuasive, because forecasting the past is not difficult. The residuals from the fitted time-series model are another name for the model's in-sample forecast errors. (LOS 2.g)
Module Quiz 2.3
1.
A The independent variable is the dependent variable lagged one period, so the model is an AR(1) model. (Module 2.2, LOS 2.d)
2.
C The first-differenced series usually does not have a unit root and is, therefore, covariance stationary. (Module 2.3, LOS 2.j)
3.
A All random-walk time series have a unit root. Time series with unit root do not have a finite mean-reverting level. (Module 2.3, LOS 2.i)
4.
A A random walk process does not have a finite mean-reverting level and hence is covariance nonstationary. An AR(1) model cannot be used to fit a covariance nonstationary time series. (Module 2.3, LOS 2.j)
5.
C For a unit root test, the null hypothesis is that the time series has a unit root. For testing for unit roots, the Dickey-Fuller (DF) test computes the conventional t-statistic, which is then compared against the revised set of critical values computed by DF. If the test statistic is significant, we reject the null hypothesis (that the time series has a unit root), implying that a unit root is not present. (Module 2.3, LOS 2.k)
Module Quiz 2.4
1.
C The standard error of the estimated autocorrelations is 1/√T, where T is the number of observations. So, if the standard error is given as 0.0632, the number of observations, T, in the time series must be (1 / 0.0632)^2 ≈ 250. (Module 2.2, LOS 2.e)
2.
A The results in the table indicate that the prediction equation is xt = 26.8625 + 0.7196·xt-1, which is estimated from an AR(1) model. (Module 2.1, LOS 2.a)
3.
LOS 2.l
A The autocorrelation in the twelfth month is not statistically different from zero (p-value: 0.5612 > 0.05). Thus, there appears to be no seasonality. (Module 2.4)
4.
A If the fourth autocorrelation of the error term differs significantly from 0, this is an indication of seasonality. (Module 2.4, LOS 2.l)
5.
C Adding an appropriate lag is an appropriate solution to seasonality. Excluding variables can sometimes be used to solve multicollinearity. Transforming using first-differencing can be a cure for nonstationarity. (Module 2.4, LOS 2.l)
6.
B The seasonal (annual) lag occurs on a quarterly basis, so the appropriate model is b0 + b1·xt-1 + b2·xt-4 + εt. The intercept b0 should be included in the model. (Module 2.4, LOS 2.l)
Module Quiz 2.5
1.
C To accurately model a time series that contains shifts, it may be necessary to strategically choose a longer or shorter sample period, or to use a first- or second-order autoregressive model. There is no accepted formula for estimating the optimal sample period (though a graphical inspection of the data may be helpful). (LOS 2.o)
2.
B ARCH is present when the variance of the error depends on the variance of previous errors. A zero autocorrelation of the error term at all lags suggests that an autoregressive model is a good fit to the data. (LOS 2.m)
3.
B If only one time series has a unit root, we should not use linear regression. If neither time series have a unit root, or if both time series have unit root and the time series are cointegrated, linear regression is appropriate to use. (LOS 2.n)
