Which of the following distributions are appropriate for modeling financial returns: normal, lognormal, stable, scale mixture of normals? Describe why or why not they are appropriate.
The statistical distribution of asset returns plays a central role in financial modeling. Assumptions on the behavior of market prices are necessary to test asset pricing theories, to build optimal portfolios by computing risk/return efficient frontiers, to value derivatives and define the hedging strategy over time, and to measure and manage financial risks [^0]. However, neither economic theory nor statistical theory exist to assess the exact distribution of returns. Distributions used in empirical and theoretical research are always the result of an assumption or estimation using data [^0].
A traditional assumption made in financial study is that the simple returns are independently and identically distributed (iid) as normal with fixed mean and variance [^1]. This modeling approach is used extensively in Modern Portfolio Theory (MPT)[^2]. In reality, it is frequently observed that returns in equity and other markets are not normally distributed. Large swings (3 to 6 standard deviations from the mean) occur in the market far more frequently than the normal distribution assumption would predict [^3].
When returns fall outside of a normal distribution, the distribution exhibits skewness or kurtosis. Skewness is known as the third "moment" [^4] of a return distribution and kurtosis is known as the fourth moment of the return distribution, with the mean and the variance being the first and second moments, respectively. Skewness and kurtosis are important because in reality few investment returns are normally distributed. It is common to predict future returns based on standard deviation, but that prediction assume a normal distribution as a model. Skewness and kurtoses measure how a measured distribution differs from a normal distribution, and as such provide an indication of the reliability of predictions based on standard deviation of a normal model [^5].
Acknowledging that few returns are normally distributed, it's possible to examine other models of distribution. For example, by transforming returns to log returns one could use a log-normal distribution. However, the log-normal assumption is not consistent with all the properties of historical stock returns. In particular, many stock returns exhibit a positive excess kurtosis [^1].
Stable distributions are capable of capturing excess kurtosis shown by historical stock returns. However, non-normal stable distributions do not have a finite variance, which is in conflict with most finance theories. In addition, statistical modeling using non-normal stable distributions is difficult [^1].
One can further use transformations and a scale mixture or finite mixture of normal distributions to model returns. Advantages of mixtures of normal include that they maintain the tractability of normal, have finite higher order moments, and can capture the excess kurtosis. Yet it is hard to estimate the mixture parameters [^1].
There is ongoing research into using techniques for examining the validity of modeling assumptions, such as extreme value theory [^0]. It would be assumed that during this course we would take a more historical approach (assuming normal or another common distribution) to learn models with a set of reduced complexities in our assumptions.
[0]: Longin, F. (2005). The choice of the distribution of asset returns: How extreme value theory can help?. Journal of Banking & Finance, 29(4), 1017-1035.
[1]: Tsay, R. S. (2014). An introduction to analysis of financial data with R. John Wiley & Sons.
[2]: Modern portfolio theory. (2016, January 6). In Wikipedia, The Free Encyclopedia. Retrieved 19:26, January 9, 2016, from https://en.wikipedia.org/w/index.php?title=Modern_portfolio_theory&oldid=698572860
[3]: Hudson, R. L. (2010). The (Mis) Behaviour of Markets: A Fractal View of Risk, Ruin and Reward. Profile Business.
[4]: Moment (mathematics). (2015, December 18). In Wikipedia, The Free Encyclopedia. Retrieved 19:18, January 9, 2016, from https://en.wikipedia.org/w/index.php?title=Moment_(mathematics)&oldid=695814896
[5]: Keating, C., & Shadwick, W. F. (2002). An introduction to omega. AIMA Newsletter.
Missed due to conflicting project rollout.
Provided three ACF diagnostic graphs for random numbers of 36, 360, and 1000 size. 1) explain the differences among the figures. Do they all indicate the data are white noise? 2) why are the critical values are different distances from the mean of zero? why are the autocorrelations different in each figure when they each refer to white noise.
Explain the differences among the figures: One thing to distinctly note is that as the sample size increase the confidence band decreases. I believe this is the only thing I can confidently say, with measurement, that is changing between the graphs (other than the obvious number and distance of autocorrelations).
Do they all indicate the data are white noise? As there doesn't appear to be an autocorrelation that exceeds the confidence interval at any lag point, it is highly likely that we're observing data that are similar to white noise.
Why are the critical values different distances from the mean of zero?: Sample size increase.
Why are the autocorrelations different in each figure when they each refer to white noise?: Differences in sample size appear to manifest in differences in the lag. We'd surmise that if the data are white noise the autocorrelations themselves would be random and not show any particular pattern.
Use R to simulate and plot some data from simple ARIMA models. Generate data from an AR(1) model with $\phi_1 = 0.6$ and $\sigma^2 = 1$ . How does the plot change as you vary $\phi$ ?
It would be important for us to use the same randomly generated data when we're sampling to create multiple time series objects. Please examine the code below to see how we sample the same generated e for each time series construction:
e = rnorm(100)
y1 = ts(numeric(100))
y2 = ts(numeric(100))
y3 = ts(numeric(100))
y4 = ts(numeric(100))
for(i in 2:100) {
y1[i] = 0.6 * y1[i-1] + e[i]
y2[i] = 0.7 * y2[i-1] + e[i]
y3[i] = 0.8 * y3[i-1] + e[i]
y4[i] = 0.9 * y4[i-1] + e[i]
}
autoplot(y1, main = expression(paste(phi, ' = 0.6')))
autoplot(y2, main = expression(paste(phi, ' = 0.7')))
autoplot(y3, main = expression(paste(phi, ' = 0.8')))
autoplot(y4, main = expression(paste(phi, ' = 0.9')))
Did some junk to get the plots in canvas (awful platform).
It appears that as
Briefly Describe what is meant by strict stationarity for a time series, and weak stationarity of order $f$ for a time series. Briefly describe the most common approaches to address nonstationarity in a time series caused by variation in the mean, and nonstationarity in a time series caused by variations in the variance.
Strict stationarity in a time series indicates that the mean and variance remain constant over the time series regardless of shift. Another way to think about strict stationarity is a time series where there exists no trend, seasonality, or cycles. The strict stationarity is akin to white noise.
Weak stationarity, of order
The most common approach in addressing nonstationarity caused by variation in the mean is to take the difference of the time series. The most common approach in addressing nonstationarity caused by variation in the variance is to perform a transform of the time series, such as logorithmic transformation.
[0]: Moment (mathematics). (2016, January 15). In Wikipedia, The Free Encyclopedia. Retrieved 01:29, February 9, 2016, from https://en.wikipedia.org/w/index.php?title=Moment_(mathematics)&oldid=700008364