This work is inspired by “ARE OUTPUT GROWTH-RATE DISTRIBUTIONS FAT-TAILED? SOME EVIDENCE FROM OECD COUNTRIES”.In their paper, they calculate growth quarter on quarter for various countries. In order to compare different countries with time series, we’ll use a proxy: industrial production. First, I had to calculate growth quarter on quarter using the equation below.
code https://github.com/grantmgeconomics93/gdp_distribution/blob/main/code.Rmd
Once we have the gdp time series, we can compare the serial correlation between ip at that time t to t -6 to see if ip can be used as a proxy for gdp.
While not a perfect match, it has a similar profile signaling that ip is applied appropriately to approximate the distribution of GDP growth over time.
Next, we can estimate a distribution IP (ARE OUTPUT GROWTH-RATE DISTRIBUTIONS FAT-TAILED? SOME EVIDENCE FROM OECD COUNTRIES) and propose the Modified distribution below.
It is a modified Laplace. A Laplace distribution is a two-sided exponential distribution.
The modified Laplace has three parameters instead of 2 for the normal Laplace. Where “The EP distribution is thus characterized by three parameters: a location parameter m, a scale parameter a, and a shape parameter b. The location parameter controls for the mean of the distribution, whereas the scale parameter is proportional to the absolute deviation”. So, I implemented a two-stage estimation strategy. First, I fitted the data to the regular two-parameter place. I did this using the fitdistr function in the MASS package. This estimates the parameters of a specified probability distribution by maximizing the likelihood of the data under that distribution. I started the optimization process at mu, equaling zero and an equaling 1. These guesses must be relatively close to the actual values for the Newton-Raphson method to improve with each estimation. Once I got estimates for the two-parameter Laplace, I could estimate the slope perimeter b using a more general optimization function creatively named optim.I will use this estimation strategy going forward. Since b is the slope parameter, it is the most important in exploring the distribution of growth.
My results for USA IP parameters.
I wanted to test the variance of b because if b had a high enough variance, the classification of Laplace could be questioned.” If b D 2, the distribution reduces to a Gaussian. If b < 2, the distribution displays tails fatter than those of a Gaussian (henceforth ‘super-Normal’ tails). If b equals 1, one recovers a Laplace. Finally, values of b smaller than one indicate tails fatter than those of a Laplace (‘super-Laplace’ tails in what follows). ”When I bootstraped the data 4000 times, I got only 3 Bs above 2 with an average B of 1.3 and a standard deviation of 0.18. This allows for high confidence that the underlying distribution is indeed Laplace. For the standard errors, I assumed normally because I was taking the mean of samples, so the central limit applies.
TESTS /CDF
I wanted to conduct goodness of fit tests. To see how well the MLE parameters that the optimization found fit the data vs how well a distribution with b equaling 1, with the other parameters being identical, To test these two series of parameters, I first had to find the CDF( cumulative density function). I created a function that integrated distribution from negative infinity to xi. We are interested in seeing whether b equals 1. If b equals 1, the distribution equals Laplace. I applied this function to the distribution with all three optimized parameters and the distribution with b equaling 1.
Once I had the two CDFs, I can do goodness of fit tests on them. The first test I tried was the Kolmogorov-Smirnov and Cramer-von Mises. The Kolmogorov-Smirnov test measures the largest distance between an empirical distribution from a chosen distribution, in this case, the empirical IP data and the distribution of custom_cdfoneUS. The p-value was 0.5669, meaning the two distributions are not significantly different. Next, I tried the Cramer-von Mises, which uses the rank and compares it to the rank of an ideal distribution.
The p-test was 0.899, Meaning it is highly likely that the Laplace with a b is close to one.
County MLEs
After I validated, the distribution of the US’s IP is close to a modified Laplace. I wanted to see if the distribution applied to other countries. I obtained the IP data for the (OECD). Then, I looped the two-stage estimation I mentioned above to see if any country had a b parameter above 2. The lowest b was .6, the highest was 1.510, and the second highest was 1.323. This backs up the thesis that the distribution is the modified Laplace mentioned above.
The countries with the highest b parameters are above. I tested if the parameters differed significantly from 1, making it a Laplace. I also tested if it differed significantly from 2, making it normal. As you can see, no country could be considered to have a normal distribution. Making the modified Laplace place a good approximation of the underlying distribution. I used Cramer Rao standard errors because the central limit does not apply. After all, I was finding the standard errors of the MLE directly.
EVOLUTION OF USA
After testing the b parameter, I want to see if the distribution was present in shorter time periods. I tried estimating the MLE in ten-year periods.
As you can see, fat tails( b<2) were still present.
Nelder-Mead and t distribution
I tried other fat tail distributions.I tried the t distribution with the pdf below.
I had to try a different estimation strategy. I had to switch to Nelder-mead.The Nelder-Mead method is a heuristic optimization technique widely used for problems where the objective function doesn’t provide derivatives or is non-smooth, noisy, or discontinuous. It operates by constructing a simplex, a polytope of n+1 n-dimensional space, which it then iteratively adjusts and shifts towards regions with lower function values. The adjustments involve reflection, expansion, contraction, and shrinking steps based on the evaluated function values at the simplex’s vertices. While it doesn’t use gradient information and can be slower to converge, especially in higher dimensions, its simplicity and robustness make it a popular choice for complex optimization problems where gradient-based methods are unsuitable. However, it may converge to a local minimum and doesn’t guarantee finding a global optimum.
The smaller the DF parameter, the fatter the tails. As you can see below, DF is primarily low, with a few exceptions. The average DF is 6.47; however, the skew is 5.7, meaning most of the values are low, suggesting fat tails. Below are the 20 highest df values.
implications
knowing the distributions of GDP distributions has a few implications. Due to the non-gaussian fat-tailed nature of growth in GDP distribution as well as “, but also for plants, firms and industries (see Stanley et al., 1996; Lee et al., 1998; Amaral et al., 1997; Bottazzi and Secchi, 2003a, 2003b; Castaldi and Dosi, 2004; Fu et al., 2005; Sapio and Thoma, 2006; among others) we should apply appropriate tests. We can also use this information for forecasting.