Time-Series and Forecasting Analytics

Introduction

Many organizational decisions depend not only on the level of a variable but also on when an observation occurs. Electricity demand varies by hour, sales by week, traffic by minute, and inflation by month. Such data are time series: observations indexed in chronological order, usually at equally spaced intervals. Time-series analytics seeks to understand temporal dependence, identify systematic structure, generate forecasts, quantify uncertainty, and support decisions such as inventory planning, staffing, capacity allocation, budgeting, and risk management.

Time-series forecasting differs from ordinary cross-sectional prediction. The order of observations matters, future values are unavailable when a model is trained, and observations are often correlated. A forecasting model must therefore be evaluated using a realistic temporal procedure rather than a random train-test split. This chapter develops a practical and statistical framework using R, covering decomposition, stationarity, exponential smoothing, ARIMA, dynamic regression, forecast evaluation, implementation, and governance.

Learning Objectives

By the end of this chapter, a learner should be able to:

1. Explain trend, seasonality, cycles, irregular variation, autocorrelation, and stationarity.

2. Explore and decompose time-series data appropriately.

3. Select transformations and differencing operations while avoiding over-differencing.

4. Construct and interpret exponential-smoothing and ARIMA models.

5. Incorporate explanatory variables through dynamic regression.

6. Evaluate forecasts using rolling-origin validation and suitable accuracy measures.

7. Implement forecasting workflows in R and communicate uncertainty and business implications.

8. Recognize limitations, ethical risks, structural breaks, and governance requirements.

1. The Nature of Time-Series Data

A time series can be represented as observations y_t recorded at times t = 1, 2, ..., T. The time index may be hourly, daily, weekly, monthly, quarterly, or annual. The frequency should be chosen according to the decision problem and the data-generating process. For example, a daily sales series is inappropriate for a monthly replenishment decision if daily values are not aggregated consistently, while monthly data may conceal operational peaks relevant to staffing.

Important terminology includes:

• Level: The typical magnitude of a series at a particular time.
• Trend: A persistent long-run increase or decrease.
• Seasonality: A recurring pattern associated with a fixed calendar or operational frequency.
• Cycle: Longer, usually less regular fluctuations around the trend.
• Irregular component: Unexplained short-term variation or noise.
• Lag: The number of time intervals separating two observations.
• Autocorrelation: Correlation between a series and its lagged values.
• Forecast horizon: The number of future periods being predicted.
• Forecast origin: The last observation available when a forecast is produced.

A useful conceptual decomposition is

y_t = T_t + S_t + E_t,

for an additive series, where T_t is the trend-cycle component, S_t is the seasonal component, and E_t is the remainder. In a multiplicative decomposition,

y_t = T_t S_t E_t.

The additive form is appropriate when seasonal fluctuations have roughly constant absolute size. The multiplicative form is more appropriate when seasonal amplitude increases with the level. A logarithmic transformation often converts multiplicative behavior into additive behavior:

log(y_t) = log(T_t) + log(S_t) + log(E_t),

provided y_t is positive.

2. Exploring Temporal Structure

A sound analysis begins with data quality and visualization. Dates must be parsed correctly, observations must be sorted chronologically, and duplicate or missing timestamps must be identified. The analyst should determine whether the sampling interval is regular. Missing periods are not automatically zero values; a missing sales record may indicate an unavailable system rather than no sales.

A time plot reveals trend, changing variance, outliers, level shifts, and possible seasonality. Seasonal plots compare observations at the same phase of a cycle, such as all January values. A subseries plot displays each seasonal position separately. These plots are often more informative than a single numerical diagnostic.

In R, a typical workflow using the forecast ecosystem is:

```r

library(forecast)

library(ggplot2)

sales_ts <- ts(sales, start = c(2019, 1), frequency = 12)

autoplot(sales_ts) +

xlab("Year") + ylab("Units sold")

ggseasonplot(sales_ts, year.labels = TRUE)

Acf(sales_ts)

Pacf(sales_ts)

```

The autocorrelation function (ACF) at lag k is

rho_k = Corr(y_t, y_{t-k}),

where Corr denotes correlation and k is a positive integer. The sample ACF is estimated from available pairs of observations. High positive autocorrelation at small lags indicates persistence. A slow decay in the ACF may indicate trend or nonstationarity rather than a meaningful long-memory process. The partial autocorrelation function (PACF) measures the correlation at lag k after accounting for intermediate lags and is useful in identifying autoregressive structure.

Outliers require investigation, not automatic deletion. A genuine promotion, strike, weather event, or cyber incident may be an important business signal. An incorrectly recorded value may be corrected. Interventions can sometimes be represented by indicator variables rather than removed.

3. Stationarity and Differencing

A weakly stationary series has a constant mean, constant variance, and autocovariance that depends only on the lag, not on calendar time. Stationarity is important because many statistical models assume that relationships estimated in the past remain stable in the near future. A series with trend or changing variance is often nonstationary.

The first difference is

Delta y_t = y_t - y_{t-1}.

It measures change between consecutive periods. A second difference is

Delta^2 y_t = Delta y_t - Delta y_{t-1}.

For seasonal period m, seasonal differencing is

Delta_m y_t = y_t - y_{t-m}.

Differencing removes certain forms of trend and seasonality, but it also removes information and can increase forecast uncertainty. The objective is not to make a series appear as random as possible; it is to obtain a plausible stationary representation with the minimum necessary differencing.

The augmented Dickey-Fuller (ADF) test evaluates a null hypothesis involving a unit root. A simplified regression is

Delta y_t = alpha + beta t + gamma y_{t-1} + sum(phi_i Delta y_{t-i}) + epsilon_t.

The number of lagged differences is selected to control residual autocorrelation. The null hypothesis is typically gamma = 0, representing a unit root. A small p-value provides evidence against the unit-root null, but the test is sensitive to specification, sample size, structural breaks, and deterministic terms. The KPSS test reverses the usual logic: its null is stationarity. Tests should therefore be combined with plots, domain knowledge, and residual diagnostics rather than treated as automatic decision rules.

A practical transformation sequence is:

1. Plot the original series and inspect variance.

2. Apply a log or Box-Cox transformation if variability rises with the level.

3. Examine whether ordinary and seasonal differencing are needed.

4. Fit candidate models and inspect residuals.

5. Avoid unnecessary differencing and check that forecasts remain interpretable.

The Box-Cox transformation is

w_t(lambda) = (y_t^lambda - 1) / lambda, if lambda is not 0,

and w_t(0) = log(y_t), if lambda = 0. It is defined for positive y_t. The parameter lambda is generally estimated from the data, but operational interpretation and back-transformation bias must be considered.

4. Exponential Smoothing and the ETS Framework

Exponential-smoothing methods generate forecasts using weighted averages in which recent observations receive greater weight. They are particularly effective when level, trend, and seasonality evolve gradually.

For simple exponential smoothing, appropriate for a series with no systematic trend or seasonality, the level update is

l_t = alpha y_t + (1 - alpha)l_{t-1},

where 0 < alpha < 1, l_t is the estimated level, and y_t is the current observation. The one-step forecast is

hat{y}_{t+1|t} = l_t.

A high alpha responds quickly to recent changes; a low alpha produces a smoother level.

Holt’s linear method incorporates trend:

l_t = alpha y_t + (1-alpha)(l_{t-1}+b_{t-1}),

b_t = beta(l_t-l_{t-1}) + (1-beta)b_{t-1},

where b_t is the trend estimate, 0 < alpha < 1, and 0 < beta* < 1. The h-step forecast is

hat{y}_{t+h|t} = l_t + h b_t.

Un-damped trends may grow unrealistically at long horizons. Damped Holt forecasting replaces h with a sum of damping powers:

hat{y}_{t+h|t} = l_t + (phi + phi^2 + ... + phi^h)b_t,

where 0 < phi < 1. The trend gradually flattens.

Seasonal methods estimate a seasonal state. In the additive seasonal form, the forecast can be expressed conceptually as level plus trend plus the relevant seasonal state. In a multiplicative form, the seasonal state scales the level. Modern exponential smoothing is commonly expressed through ETS models, classified by error type, trend type, and seasonal type. Parameters and initial states are estimated by optimizing a likelihood or an equivalent objective function.

In R:

```r

fit_ets <- ets(sales_ts)

fc_ets <- forecast(fit_ets, h = 12)

autoplot(fc_ets)

summary(fit_ets)

```

Exponential smoothing is attractive because it is adaptive, interpretable, and effective for many operational series. It does not, however, automatically explain causal drivers. A strong forecast does not prove that the estimated trend or seasonal pattern has a causal origin.

5. ARIMA Models

The ARIMA family models dependence in a stationary or differenced series. An ARIMA(p,d,q) model has p autoregressive terms, d ordinary differences, and q moving-average terms. For a stationary series x_t, an ARMA(p,q) model is

x_t = c + phi_1 x_{t-1} + ... + phi_p x_{t-p} + epsilon_t + theta_1 epsilon_{t-1} + ... + theta_q epsilon_{t-q},

where c is an intercept or constant term, phi_i and theta_j are parameters, and epsilon_t is an error term, often assumed to have mean zero and constant variance. The errors need not be normally distributed for point forecasts to be useful, but normality is commonly used for likelihood-based intervals.

For an ARIMA model, x_t is obtained after d differences of y_t. Seasonal ARIMA extends the model with seasonal autoregressive, differencing, and moving-average orders, written SARIMA(p,d,q)(P,D,Q)_m, where m is the seasonal period.

Identification uses ACF and PACF patterns as guides, not rigid rules. A rapidly decaying ACF and a sharp PACF cutoff can suggest autoregressive behavior; a sharp ACF cutoff and decaying PACF can suggest moving-average behavior. Mixed patterns often require several candidates. Information criteria such as AICc compare fitted models while penalizing complexity:

AICc = AIC + [2k(k+1)]/(n-k-1),

where k is the number of estimated parameters and n is the effective sample size. Lower AICc is preferred among models fitted to the same data and target.

R implementation is straightforward:

```r

fit_arima <- auto.arima(

sales_ts,

seasonal = TRUE,

stepwise = FALSE,

approximation = FALSE

)

fc_arima <- forecast(fit_arima, h = 12)

checkresiduals(fit_arima)

```

`auto.arima()` searches a model space using unit-root procedures, information criteria, and likelihood estimation. Its result is a useful candidate, not an unquestionable answer. The analyst should review transformations, differencing, residuals, outliers, structural breaks, and business plausibility.

Residual Diagnostics

A fitted forecasting model should leave residuals that are approximately uncorrelated, have stable variance, and have no systematic remaining structure. The residual ACF should contain no meaningful pattern. The Ljung-Box test evaluates whether a group of autocorrelations is jointly different from zero. For h tested lags, a simplified statistic is

Q = n(n+2) sum from k=1 to h of [r_k^2/(n-k)],

where n is the number of residuals and r_k is the residual autocorrelation at lag k. Degrees of freedom must be adjusted for estimated model parameters. A large p-value does not prove that residuals are white noise; it indicates insufficient evidence of residual autocorrelation under the test assumptions.

Residual diagnostics also include plots over time, a histogram or density plot, and a normal Q-Q plot where prediction intervals rely on normality. Heteroscedasticity, outliers, and volatility clustering may require alternative models or robust procedures.

6. Dynamic Regression and Transfer-Function Thinking

Forecasting can improve when external predictors are available. A dynamic regression model can be written as

y_t = beta_0 + beta_1 x_{1,t} + ... + beta_k x_{k,t} + n_t,

where n_t follows an ARIMA error process. The coefficients describe conditional associations after accounting for the error dynamics. A predictor must be known at the forecast horizon or itself forecasted. Using future information unavailable at decision time creates leakage and overstates performance.

Examples include demand predicted from price, promotions, holidays, temperature, or economic indicators. Lagged predictors can represent delayed effects, such as advertising influencing sales in subsequent weeks. Distributed-lag terms should be selected carefully to avoid multicollinearity and excessive flexibility.

In R:

```r

xreg_future <- cbind(

promotion = future_promotion,

temperature = future_temperature

)

fit_dyn <- auto.arima(

sales_ts,

xreg = cbind(promotion, temperature)

)

fc_dyn <- forecast(fit_dyn, xreg = xreg_future, h = 12)

```

The future design matrix must have exactly the required variables and horizon. If temperature is unknown, a weather forecast or scenario distribution should be used. If a promotional plan changes, the sales forecast should be recomputed. Dynamic regression therefore supports scenario analysis, but it does not eliminate uncertainty.

Correlation is not causation. A calendar indicator may forecast demand without explaining why demand changes. For causal claims, the analyst needs a research design, credible assumptions, and attention to confounding, reverse causality, and intervention timing.

7. Forecast Evaluation and Uncertainty

Forecast evaluation must imitate operational use. A random split mixes future information into the training set and is generally inappropriate for time series. Instead, use a chronological holdout or rolling-origin evaluation. In a rolling-origin design, a model is fitted using observations through time t, forecasts the next h periods, and then the origin advances. The procedure may use an expanding window or a fixed-width moving window.

Common accuracy measures include:

Mean absolute error:

MAE = (1/n) sum |y_i - hat{y}_i|.

Root mean squared error:

RMSE = sqrt[(1/n) sum (y_i - hat{y}_i)^2].

Mean absolute percentage error:

MAPE = (100/n) sum |(y_i-hat{y}_i)/y_i|.

MAE and RMSE are in the units of the target. RMSE penalizes large errors more strongly. MAPE is undefined when actual values are zero and unstable when they are close to zero. For intermittent or low-volume demand, scaled measures such as MASE are often preferable. MASE compares forecast error with the error of a naïve benchmark; values below one indicate improvement over that benchmark under the chosen scaling.

A naïve forecast uses the last observed value: hat{y}_{t+h|t}=y_t. A seasonal naïve forecast uses the observation from the corresponding previous season: hat{y}_{t+h|t}=y_{t+h-m}, where m is the seasonal period. These benchmarks are essential. A sophisticated model that cannot outperform a seasonal naïve forecast may not justify its complexity.

Prediction intervals quantify uncertainty about future observations under model assumptions. They are wider at longer horizons in many models. A point forecast is not a guaranteed outcome. Decision-makers should distinguish:

• Aleatory uncertainty: inherent future randomness.
• Parameter uncertainty: uncertainty about estimated coefficients.
• Model uncertainty: uncertainty about the selected model or structure.
• Scenario uncertainty: uncertainty about external inputs or future policies.

For positive data transformed using logs, intervals and point forecasts require careful back-transformation. Simply exponentiating the expected log forecast may underestimate the expected level because E[exp(X)] is not generally equal to exp(E[X]). Bias adjustment should be considered when the decision requires a forecast on the original scale.

8. End-to-End R Workflow

A professional workflow is reproducible and explicitly separates data preparation, model fitting, validation, forecasting, and reporting.

```r

library(forecast)

1. Create a regular time-series object after validating dates

y <- ts(units, start = c(2020, 1), frequency = 12)

2. Establish benchmarks

naive_fit <- naive(y, h = 12)

seasonal_fit <- snaive(y, h = 12)

3. Fit candidate models

ets_fit <- ets(y)

arima_fit <- auto.arima(y, seasonal = TRUE)

4. Compare using a common test period or rolling-origin design

accuracy(ets_fit)

accuracy(arima_fit)

5. Diagnose and forecast

checkresiduals(arima_fit)

future <- forecast(arima_fit, h = 12, level = c(80, 95))

autoplot(future)

```

For a formal comparison, reserve the final test period until model development is complete. Use the training portion for tuning and rolling validation, then evaluate once on the untouched test period. Refit the selected model on all available data only after the evaluation has been documented.

Reproducibility requires recording data versions, transformation choices, missing-value treatment, software versions, random seeds where relevant, forecast origin, horizon, model specification, and evaluation results. Forecast pipelines should be monitored after deployment because accuracy can deteriorate when customer behavior, policies, products, or measurement systems change.

9. Illustrative Case: Forecasting Monthly Hospital Pharmacy Demand

Background

This fictional case concerns a regional hospital that dispenses a chronic medication through its outpatient pharmacy. Monthly demand has increased over four years and exhibits strong seasonal variation. Stockouts are clinically undesirable, while excessive inventory creates expiry and working-capital costs.

Problem

The pharmacy currently orders using the previous month’s demand. Management requests a twelve-month forecast and a recommendation for a safer planning process. The data contain monthly units dispensed, a policy change that expanded eligibility, and a small number of months affected by a temporary supplier shortage.

Analysis

The analyst first validates monthly continuity and confirms that shortage months represent constrained supply rather than true demand. Treating observed dispensation as unconstrained demand would understate future requirements. The time plot shows trend and seasonal peaks. Seasonal naïve, ETS, seasonal ARIMA, and dynamic regression models are considered. The policy change is represented by an intervention indicator, while shortage months are flagged for review rather than silently deleted.

A chronological validation design uses the first 36 months for initial training and evaluates one- to three-month-ahead forecasts across later origins. MAE is used for operational interpretability, while RMSE is reported because severe underforecasting has disproportionate consequences. The forecasts are evaluated against a seasonal naïve benchmark.

Approach and Solution

The analyst fits an ETS model and a SARIMA model after examining whether a log transformation stabilizes variance. A dynamic regression model includes the eligibility-policy indicator, but the future policy status is explicitly documented. Residual diagnostics show that the initial ARIMA model retains seasonal autocorrelation. After adding a seasonal component, residual autocorrelation is no longer material according to visual inspection and a Ljung-Box diagnostic. The selected model has the lowest rolling-origin MAE, but the final inventory decision uses the upper prediction interval rather than the point forecast alone.

The pharmacy adopts a monthly review process: update the model with the latest validated data, generate a twelve-month distribution of forecasts, check policy and supply scenarios, and set reorder quantities using lead time, service-level targets, storage capacity, and expiry risk. Shortage months are separately investigated with procurement staff.

Outcome

The model provides more reliable planning than the previous-month rule, especially during recurring seasonal peaks. The organization does not treat the forecast as an automatic order quantity. Instead, it combines the forecast distribution with lead-time uncertainty and clinical service requirements. Forecast performance is monitored using rolling errors and reviewed after policy changes.

Learning Points

1. Observed demand may differ from unconstrained demand.

2. A benchmark is necessary for judging model value.

3. External interventions must be known or scenarized at the forecast horizon.

4. Prediction intervals support risk-sensitive decisions.

5. Forecasting and inventory optimization are related but distinct tasks.

10. Limitations, Governance, and Contemporary Practice

Forecast models assume that relevant patterns remain sufficiently stable. Structural breaks caused by pandemics, regulations, mergers, technology changes, or competitor entry can invalidate historical relationships. No algorithm can reliably infer an unprecedented event from ordinary historical patterns. Model ensembles, scenario analysis, expert overrides, and rapid monitoring can reduce but not eliminate this risk.

Forecasting can also create ethical concerns. A model used for hospital staffing, credit risk, policing, or social services may amplify historical inequality. A forecast is not a neutral fact when it determines resource access. Organizations should document purpose, data provenance, affected groups, validation results, override authority, and appeal mechanisms. Personally identifiable or sensitive data require appropriate access control, minimization, retention, and legal compliance.

AI and machine-learning methods, including gradient boosting, recurrent neural networks, temporal convolutional networks, and transformer-based models, can model nonlinearities and many related series. They may outperform classical models when large, clean, high-frequency datasets and stable deployment conditions exist. However, greater flexibility increases data, tuning, interpretability, and leakage risks. Classical ETS and ARIMA models remain valuable because they are transparent, statistically grounded, and strong baselines. Modern practice often combines approaches through ensembles and hierarchical or grouped forecasting.

Best practices include:

• Define the decision, horizon, frequency, and loss function before modeling.
• Establish naïve and seasonal-naïve benchmarks.
• Preserve temporal order during validation.
• Inspect data quality, missingness, outliers, and structural breaks.
• Use the minimum transformation and differencing necessary.
• Validate external predictors at the forecast origin.
• Diagnose residuals and assess interval calibration.
• Report uncertainty, not only point forecasts.
• Monitor performance after deployment and retrain when justified.
• Keep a reproducible model and decision audit trail.

Chapter Summary

Time-series forecasting analyzes observations whose chronological order creates dependence. Effective analysis distinguishes level, trend, seasonality, cycles, irregular variation, and structural change. Visualization, autocorrelation diagnostics, transformations, and carefully chosen differencing help establish an appropriate representation. Exponential smoothing provides adaptive estimates of level, trend, and seasonality, while ARIMA models represent dependence through autoregressive, differencing, and moving-average components. Dynamic regression incorporates external predictors but requires future values or scenarios and does not by itself establish causation.

Forecasts should be evaluated through chronological holdouts or rolling-origin validation against naïve benchmarks. MAE, RMSE, MAPE, and scaled measures answer different questions, and prediction intervals are essential for risk-sensitive planning. In professional settings, model selection must be connected to data quality, operational constraints, governance, monitoring, and ethical use. A reliable forecasting system is therefore not merely a fitted model; it is a reproducible decision process that recognizes uncertainty and changing conditions.

Key Terms

Time series: Chronologically ordered observations measured at regular or irregular intervals.

Trend: Persistent long-term movement in a series.

Seasonality: Repeating variation linked to a fixed calendar or operational period.

Cycle: Longer-term fluctuation without a necessarily fixed frequency.

Autocorrelation: Correlation between observations separated by a specified lag.

Stationarity: Stability of distributional properties, especially mean, variance, and autocovariance over time.

Lag: The number of time intervals between observations.

Differencing: Subtracting a lagged observation to remove trend or unit-root behavior.

Exponential smoothing: Forecasting based on weighted updating of recent observations.

ETS model: A model classified by error, trend, and seasonal components.

ARIMA: Autoregressive integrated moving-average model.

SARIMA: Seasonal extension of ARIMA.

Dynamic regression: Regression with time-dependent errors and possibly lagged predictors.

Forecast origin: Last time point available when producing a forecast.

Forecast horizon: Number of future periods predicted.

Rolling-origin evaluation: Repeated time-ordered forecast testing at successive origins.

Prediction interval: Range intended to contain a future observation with a stated probability under model assumptions.

Structural break: A persistent change in a series’ generating process.

Data leakage: Use of information unavailable at the forecast origin.

Benchmark forecast: A simple reference method used to judge model improvement.

Review Questions

Short Answer Questions

1. Distinguish trend from seasonality in a time series.

2. What does first differencing accomplish, and why should it not be applied automatically?

3. State the difference between an ACF and a PACF.

4. Why is a seasonal-naïve forecast an important benchmark?

5. What information must be available to use an external predictor in a future dynamic-regression forecast?

Long Answer Questions

1. Explain the assumptions, construction, and practical interpretation of additive and multiplicative decomposition.

2. Compare simple exponential smoothing, Holt’s method, damped trend forecasting, and seasonal ETS models.

3. Explain how ARIMA models represent autoregression, differencing, and moving-average behavior, including the role of residual diagnostics.

4. Design a rolling-origin evaluation procedure for a monthly demand-forecasting problem and justify appropriate accuracy measures.

5. Discuss the governance and ethical requirements for deploying a forecasting model in a high-impact organizational setting.

Analytical / Application Questions

1. A monthly series has increasing variance, strong annual seasonality, and an ACF that decays slowly. Propose a sequence of transformations, diagnostics, and candidate models, explaining the reasoning behind each step.

2. A retailer’s sales forecast uses price and promotion as predictors, but next quarter’s promotional plan is not finalized. Explain how this affects model use and propose a scenario-based forecasting approach.

3. A hospital observes low dispensing volumes during two months because of a supplier shortage. Explain how treating those observations as ordinary demand could bias forecasts, and design an analysis that separates demand from supply constraints.

Multiple Choice Questions with Answers

Question 1

Which condition most strongly supports the use of a multiplicative seasonal model?

A. Seasonal fluctuations have constant absolute size

B. Seasonal fluctuations increase as the series level increases

C. The series contains no repeated pattern

D. All observations are negative

Correct Answer: B. Seasonal fluctuations increase as the series level increases

Question 2

What is the purpose of first differencing a time series?

A. To introduce a deterministic trend

B. To remove every form of random variation

C. To measure changes between consecutive observations and potentially remove trend

D. To guarantee normally distributed observations

Correct Answer: C. To measure changes between consecutive observations and potentially remove trend

Question 3

In simple exponential smoothing, a larger alpha generally makes the level estimate:

A. More responsive to recent observations

B. Independent of recent observations

C. Exactly equal to the long-run mean

D. More seasonal by definition

Correct Answer: A. More responsive to recent observations

Question 4

What does the d in ARIMA(p,d,q) represent?

A. The number of seasonal cycles

B. The number of ordinary differencing operations

C. The number of external predictors

D. The number of forecast horizons

Correct Answer: B. The number of ordinary differencing operations

Question 5

Which validation method best preserves the forecasting direction of time?

A. Randomly assigning observations to folds

B. Using only the first observation for testing

C. Rolling-origin evaluation

D. Shuffling all observations before splitting

Correct Answer: C. Rolling-origin evaluation

Question 6

Which measure is especially problematic when actual values are zero?

A. MAE

B. RMSE

C. MASE

D. MAPE

Correct Answer: D. MAPE

Question 7

A residual ACF showing substantial autocorrelation most directly suggests that:

A. The model may have left temporal structure unexplained

B. The model is necessarily causal

C. The data must be cross-sectional

D. The forecast intervals are automatically calibrated

Correct Answer: A. The model may have left temporal structure unexplained

Question 8

In dynamic regression, a predictor used for a future forecast must generally be:

A. Known or forecasted for the relevant future periods

B. Measured only after the forecast period ends

C. Unrelated to the target by definition

D. Identical to the response variable

Correct Answer: A. Known or forecasted for the relevant future periods

Question 9

Which forecast is defined by using the observation from the corresponding period in the previous season?

A. Mean forecast

B. Random-walk forecast

C. Seasonal-naïve forecast

D. Zero forecast

Correct Answer: C. Seasonal-naïve forecast

Question 10

Why can simply exponentiating a forecast from a log-transformed model underestimate the expected original-scale value?

A. Because logarithms remove all seasonality

B. Because the expected exponential is generally not the exponential of the expected value

C. Because ARIMA models cannot use transformations

D. Because prediction intervals are always symmetric

Correct Answer: B. Because the expected exponential is generally not the exponential of the expected value