Time series analysis

Time-Series Analysis

A time series is a sequence of observations indexed by time, where the ordering matters and successive observations are typically dependent. That dependence violates the independence assumption behind ordinary regression, so time series needs its own toolkit. This skill covers the classical Box-Jenkins pipeline: make the series stationary, read the autocorrelation structure to identify a model, fit an ARIMA (or SARIMA) model, and validate it through residual diagnostics before forecasting.

Agent affinity: box (Box-Jenkins methodology, model identification, diagnostics), efron (computational fitting, resampling-based prediction intervals)

Concept IDs: stat-descriptive-statistics, stat-hypothesis-testing

Stationarity and Differencing

What stationarity means

A series is weakly (covariance) stationary when three things hold constant over time:

1. Constant mean: no trend drifting up or down.
2. Constant variance: spread does not grow or shrink over time.
3. Autocovariance depends only on lag: the relationship between observations k steps apart is the same everywhere in the series, not on the absolute time.

Most classical methods (AR, MA, ARMA) require stationarity. Real series usually are not stationary out of the box -- they trend, drift, or have changing variance -- so the first job is to transform them into a stationary series.

Making a series stationary

Problem	Transformation
Non-constant variance (spread grows with level)	Log or Box-Cox transform
Trend / non-constant mean	First differencing: w_t = y_t - y_{t-1}
Trend that remains after one difference	Second differencing (rarely need d > 2)
Seasonal pattern	Seasonal differencing: y_t - y_{t-s} for period s

First differencing removes a linear trend; the differenced series models the change from one period to the next. The number of regular differences needed is the d in ARIMA(p, d, q).

Testing for stationarity

• Augmented Dickey-Fuller (ADF) test: null hypothesis is a unit root (non-stationary). A small p-value is evidence for stationarity.
• KPSS test: null hypothesis is stationarity (the reverse of ADF). A small p-value is evidence against stationarity.
• Run both. Agreement (ADF rejects, KPSS does not) is strong evidence of stationarity. Disagreement usually means the series is trend-stationary or needs differencing.
• Warning against over-differencing: differencing more than necessary inflates variance and introduces artificial negative autocorrelation at lag 1. If the lag-1 autocorrelation of the differenced series is strongly negative (near -0.5) and the variance jumped, you have differenced too far.

Autocorrelation: ACF and PACF

The autocorrelation structure is the fingerprint used to identify a model.

Autocorrelation function (ACF)

The ACF at lag k is the correlation between y_t and y_{t-k}. Plotted against lag, it shows how far into the past the series "remembers." Values outside the approximate significance band +/- 1.96 / sqrt(n) are considered non-zero.

Partial autocorrelation function (PACF)

The PACF at lag k is the correlation between y_t and y_{t-k} after removing the linear effect of the intervening lags 1 ... k-1. It isolates the direct relationship at lag k.

Reading the plots to identify an order

Pattern	ACF	PACF	Suggested model
Autoregressive	Tails off (decays gradually)	Cuts off after lag p	AR(p)
Moving average	Cuts off after lag q	Tails off (decays gradually)	MA(q)
Mixed	Tails off	Tails off	ARMA(p, q)

"Cuts off" means the spikes drop abruptly into the significance band; "tails off" means they decay (geometrically or in a damped sine wave). This ACF/PACF reading is the heart of model identification.

AR, MA, ARMA, and ARIMA

Autoregressive AR(p)

y_t is a linear function of its own p past values plus white noise:

y_t = c + phi_1 * y_{t-1} + ... + phi_p * y_{t-p} + e_t

The process "regresses on itself." Stationarity requires the roots of the characteristic polynomial to lie outside the unit circle (for AR(1), |phi_1| < 1).

Moving average MA(q)

y_t is a linear function of the current and q past shocks (white-noise errors):

y_t = c + e_t + theta_1 * e_{t-1} + ... + theta_q * e_{t-q}

An MA process has memory of length exactly q -- its ACF cuts off after lag q.

ARMA(p, q)

Combines both: y_t depends on its own past values and past errors. Compact and flexible for stationary series.

ARIMA(p, d, q)

The I is "integrated": apply d regular differences to make the series stationary, then fit ARMA(p, q) to the differenced series. The three orders:

• p -- autoregressive order (from the PACF).
• d -- degree of differencing (to reach stationarity).
• q -- moving-average order (from the ACF).

ARIMA(0,1,0) is a random walk; ARIMA(0,1,1) is equivalent to simple exponential smoothing; ARIMA(0,2,2) corresponds to Holt's linear trend method.

The Box-Jenkins Methodology

Box and Jenkins formalized model building as an iterative loop rather than a one-shot fit:

1. Identification. Transform for variance (log/Box-Cox), difference for stationarity (choose d), then read the ACF and PACF of the stationary series to propose candidate (p, q) orders.
2. Estimation. Fit the candidate models by maximum likelihood (or conditional least squares) to estimate the phi and theta coefficients.
3. Diagnostic checking. Examine the residuals: they must look like white noise. If they do not, return to step 1 with a revised model.
4. Forecasting. Once a model passes diagnostics, use it to forecast and quantify uncertainty.

Model selection among candidates: prefer the model with the lowest information criterion. AIC and BIC both trade goodness-of-fit against parameter count; BIC penalizes complexity more heavily and favors smaller models. The principle of parsimony applies -- a simpler model that passes diagnostics beats a more complex one that fits marginally better in-sample. This is Box's dictum that all models are wrong but some are useful: identify a model that is useful for forecasting, not one that perfectly explains the training data.

Seasonality and SARIMA

Many series repeat on a fixed period s (12 for monthly data with an annual cycle, 4 for quarterly, 7 for daily-with-weekly). A plain ARIMA cannot capture that repetition efficiently.

The SARIMA model

SARIMA(p, d, q)(P, D, Q)_s adds a seasonal set of terms:

• (p, d, q) -- the non-seasonal (short-lag) part.
• (P, D, Q) -- the seasonal part, operating at multiples of the period s.
• s -- the seasonal period.

D is the number of seasonal differences (y_t - y_{t-s}). Seasonal AR/MA terms model correlation at lags s, 2s, 3s, etc.

Identifying seasonal orders

Read the ACF/PACF at the seasonal lags. A large spike at lag s (and 2s) in the ACF that tails off suggests a seasonal MA term; a seasonal PACF cutoff suggests a seasonal AR term -- the same "cuts off vs. tails off" logic applied at multiples of s. Combine one seasonal difference with the seasonal ACF/PACF reading to pin down (P, D, Q).

Forecasting and Prediction Intervals

Point forecasts

Once fitted, the model projects forward recursively: future values are predicted from past observed values and past estimated errors, feeding each new forecast back in. Differencing is undone (the forecasts are "integrated" back to the original scale).

Prediction intervals

A point forecast without an interval is misleading -- it hides the uncertainty. A prediction interval gives a range expected to contain the actual future value with a stated probability (e.g., 95%).

• Intervals widen as the horizon grows: forecasting 1 step ahead is far more precise than 20 steps ahead, because errors compound.
• The standard formula assumes the residuals are Gaussian white noise; the interval is roughly forecast +/- 1.96 * sigma_h, where sigma_h (the forecast standard error) grows with the horizon h.
• When the normality assumption is doubtful, bootstrap / simulation-based intervals resample the residuals to build an empirical forecast distribution -- more robust for skewed or heavy-tailed errors.
• Prediction intervals are typically too narrow in practice because they ignore parameter-estimation uncertainty and the risk that the model itself is wrong. Treat them as a lower bound on true uncertainty.

Diagnostics: Residual Analysis and the Ljung-Box Test

The core check: if the model captured all the structure, the residuals should be indistinguishable from white noise -- zero mean, constant variance, and no remaining autocorrelation.

Residual checks

Diagnostic	What it checks	Healthy result
Residual time plot	Mean and variance stability	Random scatter around zero, no trend or changing spread
Residual ACF	Leftover autocorrelation	All spikes inside the +/- 1.96/sqrt(n) band
Histogram / Q-Q plot	Normality of residuals	Roughly bell-shaped / points on the diagonal

The Ljung-Box test

Instead of eyeballing every lag of the residual ACF, the Ljung-Box test checks a group of autocorrelations jointly:

Q = n(n+2) * sum_{k=1..h} [ r_k^2 / (n - k) ]

where r_k is the residual autocorrelation at lag k and h is the number of lags tested. Under the null hypothesis "the residuals are white noise up to lag h," Q follows a chi-square distribution with h - m degrees of freedom (m = number of fitted ARMA parameters).

• Large p-value (fail to reject): good -- no evidence of leftover autocorrelation; the model is adequate.
• Small p-value (reject): bad -- structure remains in the residuals; go back to the Box-Jenkins identification step and revise p, q, or the seasonal orders.

The related Box-Pierce statistic is the older, less accurate version; Ljung-Box corrects its small-sample behavior and is the standard choice.

Common Mistakes

Mistake	Why it fails	Fix
Fitting ARMA to a non-stationary series	AR/MA theory assumes stationarity; estimates are meaningless	Difference (and/or log) first; test with ADF and KPSS
Over-differencing	Inflates variance, adds spurious lag-1 negative autocorrelation	Use the smallest d that achieves stationarity
Ignoring seasonality	A seasonal spike remains in the residual ACF	Add seasonal differencing and SARIMA (P, D, Q)_s terms
Reporting a point forecast with no interval	Hides forecast uncertainty entirely	Always attach a prediction interval; expect it to widen with horizon
Skipping residual diagnostics	An inadequate model looks fine on in-sample fit but forecasts poorly	Run the Ljung-Box test and inspect the residual ACF
Choosing the model with the best in-sample fit	Overfitting; more parameters always fit training data better	Compare with AIC/BIC and favor the parsimonious model
Evaluating forecasts on the training data	Optimistic; the model has already seen those points	Hold out the most recent observations as a test set

Cross-References

• box agent: Box-Jenkins methodology, model identification, diagnostic checking, "all models are wrong."
• efron agent: Computational fitting, bootstrap and simulation-based prediction intervals, cross-validation for time series.
• regression-modeling skill: Autocorrelated residuals (residuals-vs-order plot) are the bridge from regression into time series.
• descriptive-statistics skill: Time plots and rolling summaries precede any formal model.
• inferential-statistics skill: The Ljung-Box and Dickey-Fuller tests are hypothesis tests applied to the time domain.

References

• Box, G. E. P., Jenkins, G. M., Reinsel, G. C., & Ljung, G. M. (2015). Time Series Analysis: Forecasting and Control. 5th edition. Wiley.
• Hyndman, R. J., & Athanasopoulos, G. (2021). Forecasting: Principles and Practice. 3rd edition. OTexts.
• Shumway, R. H., & Stoffer, D. S. (2017). Time Series Analysis and Its Applications. 4th edition. Springer.
• Ljung, G. M., & Box, G. E. P. (1978). "On a measure of lack of fit in time series models." Biometrika, 65(2), 297-303.
• Brockwell, P. J., & Davis, R. A. (2016). Introduction to Time Series and Forecasting. 3rd edition. Springer.