Time Series Cheat Sheet - By Marcelo Moreno Porras - Universidad Rey Juan Carlos The Econometrics Cheat Sheet Project

Oct 13, 2026

Time Series Cheat Sheet summarizes core concepts of time series econometrics, defining stocks versus flows and point-wise versus OHLC interval structures. It outlines components such as trend, seasonality, cyclical patterns, and residual irregularities. The sheet compares static, distributed-lag, dynamic, and combined model forms, then details extended OLS assumptions for time dependence, including exogeneity, homoscedasticity, lack of autocorrelation, and asymptotic results. It also covers spurious regression causes, trend modeling, seasonal adjustment methods, and autocorrelation consequences on estimation efficiency.

LanguageEnglish

CategoryPresentations

FormatPDF

Pages3

Reading Time1 min

AccessFree to download

Basic concepts

Definitions and representations (point-in-time, aggregated, OHLC)

Components of a time series

Trend, seasonal, cyclical, residual variations

Type of time series models

Static models

Distributed-lag models

Dynamic models and combinations

Assumptions and properties

OLS model assumptions under time series

Asymptotic properties of OLS

Trends and seasonality

Spurious regression

Trend specification and extraction

Seasonality and seasonal adjustment methods

Autocorrelation

Consequences for OLS efficiency

Time Series Cheat Sheet By Marcelo Moreno Porras-Universidad Rey Juan Carlos The Econometrics Cheat Sheet Project Basic concepts Definitions Time series-succession of observations ordered in time with a fixed frequency. Given the format of a time series: ❼ Point-in-time (stock) - a single value is recorded for each period. ❼ Aggregated (flow) -values represent totals or averages over the period. ❼ Range/interval (OHLC) -each period records multiple summary statistics, such as min, max, open, close. Stochastic process-a sequence of random variables that are indexed in time. Components of a time series ❼ Trend-the long-term general movement of a series. ❼ Seasonal variations-periodic oscillations that are produced in a period equal to or inferior than a year, and can be easily identified across different years (usually the result of climatology) . ❼ Cyclical variations-periodic oscillations that are produced in a period greater than a year (are the result of the economic cycle) . ❼ Residual variations-movements that do not follow a recognizable periodic oscillation (irregular events) . Type of time series models ❼ Static models-the relation between y and x is contemporary. Conceptually: yt = β0 + β1 xt + ut ❼ Distributed-lag models-the relation between y and x is not contemporary. Conceptually: yt = β0 + β1 xt + β2 xt−1 + ··· + βs xt−(s−1) + ut The long-term cumulative effect in y when ∆x is: β1 + β2 + ··· + βs ❼ Dynamic models-lags of the dependent variable (endogeneity) . Conceptually: yt = β0 + β1 yt−1 + ··· + βsyt − s + ut ❼ Combinations of the above, like the rational distributedlag models (distributed-lag + dynamic) . Assumptions and properties OLS model assumptions under time series Under these assumptions, the OLS estimator will present good properties. Gauss-Markov assumptions extended for time series: t1 . Parameters linearity and weak dependence. a. yt must be a linear function of the β’s. b. The stochastic { (xt , yt ) : t = 1, 2 ,..., T} is stationary and weakly dependent. t2 . No perfect collinearity. ❼ There are no independent variables that are constant: Var(xj) 0 , ∀j = 1 ,..., k ❼ There is no exact linear relation between independent variables. t3 . Conditional mean zero and correlation zero. a. There are no systematic errors: E(u | x 1 ,..., xk ) = E(u) = 0 → strong exogeneity (a implies b) . b. There are no relevant variables left out of the model: Cov(xj , u) = 0, ∀j = 1 ,..., k → weak exogeneity. t4 . Homoscedasticity. The variability of the residuals is the same for any x: Var(u | x 1 , ... , xk ) = σ2u t5 . No autocorrelation. Residuals do not contain information about any other residuals: Corr(ut , us | x 1 ,..., xk ) = 0 , ∀t s t6 . Normality. Residuals are independent and identically distributed (i.i.d.): u ∼ N(0,σ2u) t7 . Data size. The number of observations available must be greater than (k + 1) parameters to estimate. (It is already satisfied under asymptotic situations) Asymptotic properties of OLS Under the econometric model assumptions and the Central Limit Theorem: ❼ Hold t1 to t3a: OLS is unbiased. E(ˆβj ) = βj ❼ Hold t1 to t3: OLS is consistent. plim(ˆβj ) = βj (to t3b left out t3a, weak exogeneity, biased but consistent) ❼ Hold t1 to t5: Asymptotic normality of OLS (then, t6 is necessarily satisfied): u ∼a N(0,σ2u) ❼ Hold t1 to t5: Unbiased estimate of σ2u . E(σˆ2u) = σ2u ❼ Hold t1 to t5: OLS is BLUE (Best Linear Unbiased Estimator) or efficient. ❼ Hold t1 to t6: Hypothesis testing and confidence intervals can be done reliably. Trends and seasonality Spurious regression-is when the relation between y and x is due to factors that affect y and have a correlation with x, Corr(xj , u) 0. Is the non-fulfilment of t3 . Trends Two time series can have the same (or contrary) trend, which should lead to a high level of correlation. This can provoke a false appearance of causality; the problem is

FAQs

What is a time series, and how are stock, flow, and OHLC formats different?

A time series is a succession of observations ordered in time with a fixed frequency. Point-in-time (stock) records a single value per period; aggregated (flow) stores totals or averages; OHLC records multiple summary statistics such as min, max, open, and close.

What distinguishes static, distributed-lag, and dynamic time series models?

Static models relate y to x contemporaneously. Distributed-lag models relate y to current and past x values, capturing cumulative long-term effects. Dynamic models include lags of the dependent variable, addressing endogeneity, and combinations can mix both approaches.

Why does spurious regression happen, and how can adding a trend help?

Spurious regression occurs when an apparent relation between y and x is driven by factors affecting y that correlate with x, meaning key assumptions fail. Including a trend variable in the model can reduce the false appearance of causality, and trend can also be extracted using the Hodrick-Prescott filter.

Oct 13, 2026

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