Variable Slope Forecasting Methods and COVID-19 Risk

Abstract

There are many real-world situations in which complex interacting forces are best described by a series of equations. Traditional regression approaches to these situations involve modeling and estimating each individual equation (producing estimates of “partial derivatives”) and then solving the entire system for reduced form relationships (“total derivatives”). We examine three estimation methods that produce “total derivative estimates” without having to model and estimate each separate equation. These methods produce a unique total derivative estimate for every observation, where the differences in these estimates are produced by omitted variables. A plot of these estimates over time shows how the estimated relationship has evolved over time due to omitted variables. A moving 95% confidence interval (constructed like a moving average) means that there is only a five percent chance that the next total derivative would lie outside that confidence interval if the recent variability of omitted variables does not increase. Simulations show that two of these methods produce much less error than ignoring the omitted variables problem does when the importance of omitted variables noticeably exceeds random error. In an example, the spread rate of COVID-19 is estimated for Brazil, Europe, South Africa, the UK, and the USA.

How US Fiscal and Monetary Policy affect the GDP of Countries with Fixed and Flexible Exchange Rates: Estimates using Korean Data from 1963 to 2022

“…Aitken (1935) implies that the GLS estimate of α 0 will be the Best Linear Unbiased Estimate (BLUE) if the q i s are i.i.d. N(μ q , σ q 2 ) because GLS is BLUE for heteroscedastic models and Leightner et al (2021) show that equation ( 7) is a heteroscedastic model if q i is unknown. Leightner et al (2021) test the three methods using simulations (Leightner 2015 also provides simulation tests but solely for RTPLS).…”

“…This implies that the accuracy of calculating a separate slope estimate for each observation using equation ( 12) depends primarily upon the accuracy of the α 0 estimate. 7) and (8) (13) Leightner et al (2021) explore three ways to obtain an estimate for α 0 : they are (i) using…”

“…12) rearranged ( 14) Leightner et al (2021) show that when the omitted variable problem is ignored by estimating equation (5) using OLS, the resulting estimate for β 1 is approximately α 1 + α 2 E[q t ] which leaves an "error" for the t = i th observation of approximately α 2 X i (q i -E[q i ]) + u i . The three methods of correcting for the omitted variables problem explored by Leightner et al (2021) would be better…”

“…Leightner et al (2021) test the three methods using simulations (Leightner 2015 also provides simulation tests but solely for RTPLS). Leightner et al (2021) run sets of 5,000 simulations each for the 27 combinations of 100 observations, 250 observations, and 500 observations with the omitted variable making a 1,000 percent difference to the slope, a 100 percent difference to the slope, and a 10 percent difference to the slope, and with random error being zero, one percent, and ten percent. Leightner et al (2021) gives the name "Variable Slope Ordinary Least…”

“…Cogan et al (2010) using the same data found very different government spending multipliers when a Romer-Bernstein Keynesian model is used versus a Taylor Keynesian model. Fortunately, Leightner (2015) and Leightner, et al (2021) explain Reiterative Truncated Projected Least Squares (RTPLS) -a solution to the omitted variables problem of regression analysis which makes estimations possible without having to develop and justify complete macroeconomic models. Applying RTPLS to quarterly data from 1962 to 1997 (during which time the Republic of Korea had a fixed exchange rate regime) produces estimates where increases in US government spending decreased Korea's GDP while increases in US money supply increased Korea's GDP.…”