ECONMID

Data on multiple entities at one point in time; example: income of 100 households in 2026.

Data on one entity across multiple time periods; example: monthly U.S. unemployment.

Data tracking multiple entities across multiple time periods; example: annual sales for 30 firms over 8 years.

A numerical summary of a random outcome, such as earnings of recent college graduates.

The average of a variable across the entire population.

The average of observations in a sample; used to estimate the population mean.

A measure of the sampling uncertainty or precision of an estimate; a smaller standard error generally means greater precision.

The baseline claim being tested, often that a coefficient or autocorrelation equals zero.

The competing claim against the null hypothesis.

Assuming the null hypothesis is true, the probability of observing a result at least as extreme as the sample result.

If p-value < 0.05, reject H0 at the 5% significance level. If p-value >= 0.05, fail to reject H0.

There is insufficient evidence to reject the null hypothesis; this does not prove the null hypothesis is true.

An estimate measured in standard-error units; for testing whether a coefficient is zero, t = estimated coefficient / standard error.

A range of plausible values for a population parameter at a specified confidence level.

Measures the strength and direction of the linear relationship between two variables.

Correlation coefficient r > 0; variables tend to move in the same direction.

Correlation coefficient r < 0; variables tend to move in opposite directions.

A correlation close to +1 or -1 indicates a strong linear relationship; a correlation near 0 indicates a weak linear relationship.

Two variables may move together without one causing the other.

A model with one dependent variable and one explanatory variable.

The outcome being explained or forecast; in the R formula y ~ x, y is the dependent variable.

A variable used to explain variation in the dependent variable; in y ~ x, x is the explanatory variable.

The predicted value of y when x = 0 in a linear regression.

The estimated change in predicted y associated with a one-unit increase in x in simple linear regression.

Estimates regression coefficients by minimizing the sum of squared residuals.

Actual value minus fitted value: e = y - y_hat.

Measures how well the regression line fits the data; in standard OLS regression with an intercept, it ranges from 0 to 1.

Regression that uses more than one explanatory variable to explain a dependent variable.

Imports a dataset into a data frame.

Applies a function, such as mean or standard deviation, across multiple elements or columns.

Provides descriptive summaries, including minimum, quartiles, median, mean, and maximum for numeric data.

Creates a plot, such as a scatterplot or time-series plot, depending on the inputs.

Adds a title to the current plot.

Calculates the sample correlation between x and y.

Fits an OLS linear regression with y as the dependent variable and x as the explanatory variable.

Displays regression output such as estimated coefficients, standard errors, t-statistics, p-values, and R-squared.

Adds a fitted regression line from a model to the current plot.

A long-term systematic change or directional movement that does not appear to be periodic.

A repeating pattern linked to a fixed natural period, such as months, quarters, or days of the week.

Repeating fluctuations that do not correspond to a fixed calendar period, such as economic business cycles.

The irregular, unsystematic remainder after trend and seasonality are removed.

A trend that follows a predictable path over time.

A trend whose trajectory changes unpredictably over time.

Seasonality repeats at a fixed period on the calendar; cycles recur without a fixed calendar period.

x_t = m_t + s_t + z_t. Observed series = trend + seasonal component + random component.

When the magnitude of seasonal variation does not depend on the trend level and stays roughly constant.

x_t = m_t * s_t * z_t. Observed series = trend * seasonal component * random component.

When seasonal variation grows or shrinks proportionally with the trend level.

Additive combines components by addition; multiplicative combines them by multiplication. Constant seasonal size suggests additive; proportional seasonal size suggests multiplicative.

Subtract the estimated seasonal component: adjusted value = observed value - estimated seasonal component.

Divide by the estimated seasonal factor: adjusted value = observed value / estimated seasonal factor.

Observed value = 150 and seasonal component = 20. Adjusted value = 150 - 20 = 130.

Observed value = 150 and seasonal factor = 1.20. Adjusted value = 150 / 1.20 = 125.

Smoothing short-term fluctuations to estimate the underlying trend.

To align a moving-average trend estimate with the appropriate time period, especially when the moving-average window has an even number of observations.

m_hat_t = [0.5*x_(t-6) + x_(t-5) + ... + x_t + ... + x_(t+5) + 0.5*x_(t+6)] / 12.

The two endpoint observations each receive weight 1/24; each interior observation receives weight 1/12. The weights sum to 1.

A 12-month average falls halfway between months; averaging two overlapping 12-month averages aligns the estimate with an integer month.

Creates a time-series object.

Use frequency = 12 when creating a monthly ts object.

Use frequency = 4 when creating a quarterly ts object.

Inspect the object type, start period, end period, and observations per cycle of a time series.

Aggregates data to a coarser frequency, such as converting monthly data to annual data.

Compares the distribution of observations across seasonal positions, such as months, to help identify seasonal peaks and troughs.

Performs additive time-series decomposition into trend, seasonal, and random components.

Performs multiplicative time-series decomposition into trend, seasonal, and random components.

Combines time series while keeping only dates for which all series have observations.

Combines time series while keeping all dates and filling unavailable observations with NA.

The number of time periods ahead being forecast.

Short-term forecasts are generally more accurate than long-term forecasts.

Understanding the causes of the variable, having substantial relevant data, and forecasting a variable that is not strongly affected by the forecast itself.

A related variable whose changes reliably precede changes in the target by one or more time periods.

A long lead time and a close relationship with the target series.

A related variable that moves contemporaneously with the target but is easier or faster to forecast.

The mean of the time series remains constant over time.

The variance of the time series remains constant over time.

Mean stationarity means the average does not change over time; variance stationarity means the variability does not change over time.

The correlation of a variable with itself at different points in time, separated by a lag.

The number of time steps separating two observations.

Measures autocorrelation at different lags of a time series.

A plot of sample autocorrelation against lag; the x-axis is lag and the y-axis is autocorrelation.

Approximate 95% confidence bounds for autocorrelation under the null of no autocorrelation at that lag.

Reject H0: rho_k = 0 at the 5% significance level for that lag.

Large positive autocorrelations at early lags that decay slowly toward zero. This does not reveal whether the underlying trend is upward or downward.

Prominent spikes at multiples of the seasonal frequency; monthly annual seasonality can produce spikes at lags 12, 24, and 36.

A series whose autocorrelation spikes at nonzero lags stay within the 95% confidence bounds, indicating no detected serial dependence.

Reject the null of no autocorrelation across the series if more than 5% of the spikes fall outside the confidence interval.

Plots or computes the sample autocorrelation function for a time series.

Computes the ACF while allowing missing values to pass through, such as in some decomposition residual series.

Plots the cross-correlation function between two series across positive and negative lags.

Plots multiple time series on the same axes.

Examining relationships between two time series at different relative lags, including whether one series may lead the other.

OLS commonly assumes residuals are uncorrelated; remaining serial correlation can violate this assumption and make prediction intervals larger.

No. Residuals may still have autocorrelation after detrending and seasonal adjustment.

A smoothing method that assigns greater weight to recent observations and exponentially decreasing weights to older observations.

a_t = alpha*x_t + (1 - alpha)*a_(t-1).

The smoothing parameter, where 0 < alpha < 1, that controls the weight placed on the latest observation.

The newest observation receives most of the weight; smoothing is low and the estimate responds quickly.

The previous smoothed level receives most of the weight; smoothing is high and the estimate responds slowly.

Their weights decrease exponentially as their lag increases.

a_t = alpha*x_t + alpha*(1-alpha)*x_(t-1) + alpha*(1-alpha)^2*x_(t-2) + ...

x_hat_(n+k|n) = a_n for every horizon k >= 1; the forecast is flat at the latest smoothed level.

Choose alpha to minimize the sum of squared one-step prediction errors (SS1PE).

SS1PE = sum of (x_t - a_(t-1))^2 over the observations used for one-step predictions.

alpha = 0.30, x_t = 120, and a_(t-1) = 100. Then a_t = 0.30*120 + 0.70*100 = 36 + 70 = 106.

106, because simple exponential smoothing produces a flat forecast for every future horizon.

An extension of exponential smoothing that models level, trend, and seasonality, allowing these components to change over time.

The estimated underlying level of the series; in the additive formulation, it is seasonally adjusted.

The estimated slope, or change in the underlying level from one period to the next.

The estimated seasonal effect for a particular seasonal position.

Smoothing of the level.

Smoothing of the trend or slope.

Smoothing of the seasonal component.

a_t = alpha*(x_t - s_(t-p)) + (1-alpha)*(a_(t-1) + b_(t-1)).

b_t = beta*(a_t - a_(t-1)) + (1-beta)*b_(t-1).

s_t = gamma*(x_t - a_t) + (1-gamma)*s_(t-p).

x_hat_(n+k|n) = a_n + k*b_n + s_(n+k-p).

When the seasonal effect is approximately constant in size as the series level changes.

a_t = alpha*(x_t / s_(t-p)) + (1-alpha)*(a_(t-1) + b_(t-1)).

b_t = beta*(a_t - a_(t-1)) + (1-beta)*b_(t-1).

s_t = gamma*(x_t / a_t) + (1-gamma)*s_(t-p).

x_hat_(n+k|n) = (a_n + k*b_n)*s_(n+k-p).

When seasonal variation grows or shrinks proportionally with the series level.

Additive forecasts add the seasonal effect; multiplicative forecasts multiply by the seasonal factor. Use additive for roughly constant seasonal size and multiplicative for proportional seasonal size.

a_n = 100, b_n = 5, k = 2, and seasonal effect = 12. Forecast = 100 + 2*5 + 12 = 122.

a_n = 100, b_n = 5, k = 2, and seasonal factor = 1.20. Forecast = (100 + 2*5)*1.20 = 132.

Fits simple exponential smoothing by disabling trend and seasonal smoothing in the HoltWinters function.

Fits an additive Holt-Winters model with trend and seasonality.

Fits a multiplicative Holt-Winters model.

Generates h-step-ahead forecasts from a fitted Holt-Winters model.

A sequence of independent, identically distributed random variables with mean zero, constant variance sigma squared, and zero autocorrelation at all nonzero lags.

Independent and identically distributed: observations are independent and share the same probability distribution.

Mean zero, constant variance, and zero autocorrelation at all nonzero lags; the study guide also specifies independent, identically distributed observations.

x_t = x_(t-1) + w_t, where w_t is a white-noise shock.

The current value equals the previous value plus a new unpredictable shock.

x_t = w_t + w_(t-1) + w_(t-2) + ... plus the initial value as appropriate; the level accumulates past shocks.

B*x_t = x_(t-1); it represents a one-period lag.

x_t = B*x_t + w_t.

Generally no. The random-walk level is nonstationary in mean and variance according to the study guide.

Delta x_t = x_t - x_(t-1); it measures the change from one period to the next.

Delta x_t = x_t - x_(t-1) = w_t; the first difference equals the random-walk shock.

If its shocks are white noise, the differenced series is white noise and is stationary.

x_t = x_(t-1) + delta + w_t, where delta is the constant drift term.

White noise fluctuates around its mean without serial dependence; a random walk accumulates shocks, so its level is generally nonstationary.

A deterministic trend follows a predictable path; a random walk has a stochastic trend that changes unpredictably due to shocks.

The model may have captured systematic patterns such as trend and seasonality, but the residual ACF should also be checked for remaining autocorrelation.