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Compute the estimated trend value of a time series, and from it construct an interval forecast.

FRM® Part I · Quantitative Analysis · Non-Stationary Time Series · FRM1.QA.11.g

The idea

Once a trend model has been estimated, the estimated trend value for any date is found by inserting the time index. For the linear model, ŷ = b0 + b1 × t. For the log-linear model, ŷ = exp(b0 + b1 × t). A point forecast h steps ahead uses t = T + h. An interval forecast is the point forecast plus or minus a multiple of the standard error of the regression. That standard error is the residual standard deviation. Standard error = √(RSS ÷ (n − k)). Here k is the number of estimated parameters. The multiple is about 1.96 for 95%, 1.645 for 90% and 2.576 for 99%. This ignores parameter uncertainty, and the width does not grow with h. For a log-linear model, build the interval on the log scale first, then exponentiate both endpoints, which gives an asymmetric interval. The usual mistake is using the standard deviation of y itself.

Formulas

Interval forecast around a trend forecast

interval = ŷ ± z × SE

ŷ
the point forecast from the trend model
z
the multiple, about 1.96 for 95%, 1.645 for 90%, 2.576 for 99%
SE
the standard error of the regression, the residual standard deviation

SE is the residual standard deviation, not the standard deviation of y itself. For a log-linear model build the interval in logs and exponentiate both ends.

Standard error of a trend regression

SE = √(RSS ÷ (n − k))

RSS
the residual sum of squares
n
the number of observations
k
the number of estimated parameters, intercept included

k counts every estimated parameter, so a model with an intercept, a trend and three seasonal dummies has k = 5.

A practice question

From the question bank, with the reason each wrong option is wrong.

A trend model has been estimated by OLS and the point forecast for a future date is available. How is a 95% interval forecast constructed?

  1. Point forecast plus or minus 1.96 times the standard deviation of the series y itselfWhy not: Uses the unconditional standard deviation of y, which includes the trend variation; the residual standard error is relevant.
  2. Point forecast plus or minus 1.96 times the variance of the residualsWhy not: Uses the variance rather than the standard deviation.
  3. Point forecast plus or minus 1.96 times the residual standard error times the square root of hWhy not: Applies the random-walk scaling; in a trend model the error does not accumulate with the horizon.
  4. Point forecast plus or minus 1.96 times the standard error of the regression - correct

The error around a trend line is the residual, so the width uses the residual standard deviation: forecast +/- z sigma-hat. It ignores parameter uncertainty.

A harder one

A linear trend fitted to 80 observations gives y-hat_t = 12.0 + 0.35 t, with a standard error of the regression of 3.2. What is the 95% interval forecast for y at 4 steps ahead (t = 84)?

  1. 33.7 to 46.3Why not: Evaluates the trend at t = 80, the last observation, instead of t = 84.
  2. 35.1 to 47.7 - correct
  3. 36.1 to 46.7Why not: Uses the 90% multiplier of 1.645 for a 95% interval.
  4. 21.3 to 61.5Why not: Multiplies by the variance of the residuals (10.24) instead of the standard error.

Point forecast = 12.0 + 0.35 x 84 = 41.4. Interval = 41.4 +/- 1.96 x 3.2 = 41.4 +/- 6.27, i.e. 35.1 to 47.7.

The hardest tier

A log-linear trend fitted to 100 observations gives ln(y-hat_t) = 2.1 + 0.02 t, with a standard error of the regression of 0.06 (in log units). What is the 95% interval forecast for y at t = 105?

  1. 66.6 to 66.8Why not: Adds the log-unit standard error to the exponentiated forecast instead of working on the log scale.
  2. 53.6 to 67.9Why not: Evaluates the trend at t = 100 instead of t = 105.
  3. 4.08 to 4.32Why not: Reports the interval on the log scale without exponentiating the endpoints.
  4. 59.3 to 75.0 - correct

ln forecast = 2.1 + 0.02 x 105 = 4.20. Log interval = 4.20 +/- 1.96 x 0.06 = 4.0824 to 4.3176. Exponentiating gives 59.3 to 75.0, around 66.7.

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