Current Midyear Municipal Budget Forecast Accuracy

Abstract

This chapter examines current year budget forecast performance for municipal governments in the United States. Midyear forecasts are especially important for governments because budgets authorize public spending and changes to legal financial plans. Ample research examines forecasts beyond the current fiscal year, yet little work analyzes forecasts made during the current fiscal year for the remainder of that year. Although an examination of the literature suggests that midyear forecasts can reflect substantial improvement, the empirical analysis shows no significant evidence of improvement and even finds evidence that expenditure forecasts become worse at shorter time horizons.

Appendix: Equations and Discussion of Theil’s U

All equations used in this chapter are included in this appendix.

$$ \mathrm{APE}=\left|\frac{\mathrm{Actual}-\mathrm{Forecast}}{\mathrm{Actual}}\right| $$

(13.1)

$$ \mathrm{MAPE}=\frac{\sum \mathrm{APE}}{n} $$

(13.2)

$$ \mathrm{SAPE}=\left|\frac{\mathrm{Actual}-\mathrm{Forecast}}{\raisebox{1ex}{$\left(\mathrm{Actual}+\mathrm{Forecast}\right)$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right| $$

(13.3)

$$ \mathrm{SMAPE}=\frac{\sum \mathrm{SAPE}}{n} $$

(13.4)

$$ {U}_2=\frac{{\left[\sum \limits_{i=1}^n{\left({P}_i-{A}_i\right)}^2\right]}^{\frac{1}{2}}}{{\left[\sum \limits_{i=1}^n{A_i}^2\right]}^{\frac{1}{2}}} $$

(13.5)

For this Theil’s U formula, P = the forecasted change and A = the actual change (Bliemel 1973). For all other equations, the actual and forecast refer to the whole numbers.

$$ \mathrm{PCT}\kern0.5em \mathrm{Improved}=\frac{\mathrm{M}\kern0.5em \mathrm{Adopted}-\mathrm{M}\kern0.5em \mathrm{Modified}}{\mathrm{M}\kern0.5em \mathrm{Adopted}} $$

(13.6)

where M refers to an error measure (MAPE, SMAPE or Theil’s U₂).

$$ \mathrm{PCTBTTR}=\frac{\mathrm{SAPE}\kern0.33em \mathrm{Adopted}-\mathrm{SAPE}\kern0.33em \mathrm{Modified}}{\mathrm{SAPE}\kern0.33em \mathrm{Adopted}} $$

(13.7)

Alternate calculations of Theil’s U not used:

$$ {U}_1=\frac{{\left[\frac{1}{n}\sum \limits_{i=1}^n{\left({P}_i-{A}_i\right)}^2\right]}^{\frac{1}{2}}}{{\left[\frac{1}{n}\sum \limits_{i=1}^n{A_i}^2\right]}^{\frac{1}{2}}+{\left[\frac{1}{n}\sum \limits_{i=1}^n{P_i}^2\right]}^{\frac{1}{2}}} $$

(13.8)

This equation has two variants, where P and A may either refer to the whole numbers (forecast and actual) or the forecasted and actual changes (Bliemel 1973). Zakaria and Ali (2010) offer an apparent clarification of U₁ incorporating the change formulation as:

$$ {U}_3=\frac{{\left[\frac{1}{t}\sum \limits_{i=1}^t{\left[\left({P}_i-{A}_{i-1}\right)=\left({A}_i-{A}_{i-1}\right)\right]}^2\right]}^{\frac{1}{2}}}{{\left[\frac{1}{t}\sum \limits_{i=1}^t\kern0.28em {\left({P}_i-{A}_{i-1}\right)}^2\right]}^{\frac{1}{2}}+{\left[\frac{1}{t}\sum \limits_{i=1}^t\kern0.28em {\left({A}_i-{A}_{i-1}\right)}^2\right]}^{\frac{1}{2}}} $$

(13.9)

It is also possible that U₂ can be calculated in both variants (whole number or change). Bliemel (1973) finds that U₁ fails to effectively distinguish between accurate and inaccurate forecast, but U₂ does. U₁ and U₂ can be confused, as seen with this internet resource⁹ that reverses the labels between U₁ and U₂.

Dean (1976) reintroduces the use of mean squared values in the place of squared values for U₂ and gives an alternate equation as¹⁰:

$$ {U}_2=\frac{{\left[\frac{1}{n}\sum \limits_{i=1}^n{\left({P}_{i+1}-{A}_i\right)}^2\right]}^{\frac{1}{2}}}{{\left[\frac{1}{n}\sum \limits_{i=1}^n{A}_i^2\right]}^{\frac{1}{2}}} $$

(13.10)

Carabotta (2014) gives the equation (although labeled “T,” it reflects the U statistic development):

$$ T=\frac{RMS{E}_{e_t}}{RMS{E}_{AR(1)}} $$

(13.11)

Next, an internet source¹¹ uses proportionate values of change and calculates U₂ as¹²:

$$ {U}_2=\frac{{\left[\frac{1}{T}\sum \limits_{i=1}^{T-1}\frac{{\left({P}_{i+1}-{A}_{i+1}\right)}^2}{A_i}\right]}^{\frac{1}{2}}}{{\left[\frac{1}{T}\sum \limits_{i=1}^{Tn}{\left(\frac{A_{\left(i+1\right)}-{A}_i}{Ai}\right)}^2\right]}^{\frac{1}{2}}} $$

(13.12)