Forecasting Insurance Premiums
Producer Price Index by Industry: Premiums for Property and Casualty Insurance (Commercial Auto Insurance):
We begin with taking out trends and eliminating seasonality from the data, by running regressions and saving the residuals to forecast. The data I used was the CPI for commercial auto insurance premiums from June 1998 to April 2017, and after creating a correlogram of the index after eliminating seasonality, there were still variables affecting the data. The objective is to run enough regressions to create a residual we are able to forecast.
I used two moving average (MA) models, two autoregressive (AR) models, and an autoregressive moving average (ARMA) model. From this, we used the sample autocorrelations and observed which model had the best results in comparison to r squared, adjusted r squared, AIC and SIC values.
We compare the trend models with SIC and AIC in addition to R squared, because forecasting models solely dependent of the the highest R squared value are not the most accurate. The mean error adjusted for degrees of freedom is the adjusted r squared value, which minimizes the standard error of regression. Two other approximations are AIC and SIC. AIC penalizes degrees of freedom more heavily than adjusted r squared, but is inconsistent, whereas SIC is consistent. But, AIC is asymptotic efficient, meaning even when the sample size grows, the forecast error variances approach that of a true model with the same parameters. With this, we selected the criteria that would best forecast our own data. I used ARMA which had the best approximations from each criterion.
I forecasted the original index two times, the second a forecast over a different time period. The data I chose to forecast had an unpredictable curve that the program could not foresee. I took the trend from June 1998 to March 2013, instead of to June 2017. And it generated a smaller, more accurate interval. Here, I developed a positive and negative trend to create the interval around the curve.