*In the previous article, a statistical analysis between rare metals considering four periods of time confirmed the values estimated by the maximization for our parameter theta are equal to (-1/256) in the Gold columns. Consequently, the maximum correlation with the other rare metals lagged at (t – (1/256) (a one time lag), is reached by the time series Gold. It anticipated the other rare metals, thus becoming the most significant asset according to our analysis. We are now going to perform a test that will enlighten us on the independency of the log returns and the normality distribution over the periods previously considered. But first, a little overture about the history and properties of Gold.*

**Overview on Gold:**

For thousands of years, gold has been considered valuable. It’s not common, because rocks are not valuable, e.g. not all rocks are valuable, but certain kind of rocks like gold, silver and diamonds have value because they are rare. When you introduce something that everybody finds valuable, like a precious stone, it can hold its value for longer.

Gold is rare, but not so rare that no one can get it, it’s a perfect balance. If we look at the periodic table, we see that we can hold that. If we consider oxygen or mercury we can’t hold it, they are not practical. Gold is good also because it’s not radioactive or poisonous. It has no risk, it’s relatively safe because it can also be eaten.

Gold is considered a coin metal, and if you own coins, you don’t want that their value disintegrates before you use them. It has a good melting point compared to other materials, like titanium. Being reflective, shiny, it has stimulated the human desire for that.

How can we find gold? The process is not easy, but long and expensive. The first step is called “__Exploration__”, where geology, geography, chemistry and engineering experts work together to detect the presence of gold. The procedure requires around 10 years, and on average only 10% of global gold deposit contain sufficient gold to mine. The second stage is the “__Development__” phase, which can last from 1 to 5 years. This is the planning and construction phase of a mining process. Permits and licenses are mandatory which requires several years. We can distinguish two kinds of mines, the “surface mine” and the “underground mines”. Then “__Operation__” passage consists in extracting and processing gold. The last step is “__Decommissioning and closure__”.

How can we use gold? It has been used for different reasons, the most important of which are: jewellery, investment, technology and Central Banks. It plays a dominant role in the jewellery market, due to its intrinsic value, its beauty and its unique resistance to tarnish and corrosion. It enhances risk management and capital preservation for institutional and private investors. Technology division includes use of gold in electronics, computers, dentistry, medicine, aerospace, etc. Central Banks buy gold for long-term strategies, like hedge against policy actions. They don’t focus on the current price of their investment, instead they focus on diversification.

We can see by chart 4 that in 2016 the 47% of the gold has been required for the jewellery sector, the 36% for investment and only the 9% and 8% for Central Banks and for Technology sector. Furthermore, looking at the past three year, the amount of gold required has been more or less stable during this time. Chart 5, which values are expressed in tonnes of gold, can help us to understand that the demand for gold in the last three years for technology has not changed, that one for jewellery and Central Banks had decreased, while the only one that has been growing exponentially is the demand for investments.

Looking at the Supply side, during the last three year, in chart 6, we can distinguish three main areas, which are mine production, net producer hedging and recycled gold. Analysing the supply of gold, we can see that there is no relevant variation in volumes.

It is important to stress, in table 4 that for each year, the difference between gold supply and gold demand has always generated a surplus, which has reached its highest peak in the last year.

It is curious to consider the consumers’ demand of gold for the main countries and the world. This result is given by the sum of jewellery demand, and the total bar and coin demand in these countries. From the table 5 and chart 7 correlated to the consumer demand in the selected countries it is possible to draw certain conclusions. First, we can see that the world demand of gold raises till 2013 to gradually fall after that year. Second, if we consider the one for the main countries, every country’s demand falls after 2013, but comparing the demand since the beginning of the selected period (2010-2016), and using ln-returns, the only countries that increased their demand were the Asian one, like China, Hong Kong, Taiwan, while all the others reduced their requests.

We can observe that there is a decrease in the percentage change consumer gold demand for almost every country, with the exception of China and USA (where it stays somewhat stable). Though it is subjected to significant variations, especially in eastern countries, the general trend is not positive. We can only assume that gold, despite being a precious metal and a valuable trade market item, isn’t gifted with much suitability and desirability from the majority of countries, them preferring other forms of trade market items.

**Statistical analysis on Gold:**

Now we will test whether the Gold log-returns follow a normal distribution. Chart 8 above shows a graphical analysis for both periods. To verify the log-normal distribution of the returns, a computing of the log-return is required, and we must compare its density with a theoretical density of a normal distribution. The last step will consist in performing a hypothesis test for normality. The log-normal properties can be used to provide information about the probability distribution of the continuously compounded rate of return of the stock between time 0 and T.

The density of out-data (red line) is compared with a theoretical Gaussian distribution (blue line) with parameters equal to the real one computed on log returns of Gold in chart 10. As it is shown from chart 9 and 10, our data doesn’t follow a normal distribution in fact the theoretical distribution is quite far from the reality. The real density is peaked than the theoretical one, and through the computing of the kurtosis it is obvious that the value confirms this fact, for we have values distant from the kurtosis of a Gaussian distribution with the value of three. The only exception seems to happen during the second period, from April 2014 to April 2015. However, we need to perform other tests to be sure of that.

The Quantile-Quantile Plot test graphically must confirm if the log-returns follow a normal distribution. In this case the Q-Q Plot, namely the black line which graphically represents the quantile of the distribution of log-returns, doesn’t fall on the blue straight line, which instead represents the theoretical quantile of a normal distribution. The advantages of the q-q plot are the following: a) The sample sizes do not need to be equal; b) Many distributional aspects can be simultaneously tested. For example, shifts in location, shifts in scale, changes in symmetry, and the presence of outliers can all be detected from this plot. If the two data sets come from populations whose distributions differ only by a shift in location, the points should lie along a straight line that is displaced either up or down from the 45-degree reference line.

As we can see from chart 11, again we have a clear signal that our data do not follow a normal distribution. This is because the empirical quantiles (black points) do not fit the blue line representing the theoretical quantiles of a normal distribution. The only doubt is always for the second period.

In order to confirm definitely the normality/non-normality of the distribution of log return, we now perform the Jarque-Bera Test and the Shapiro-Wilk normality test.

The Shapiro-Wilk test utilizes the null hypothesis principle to check whether a sample x_{1}, …, x_{n} came from a normally distributed population. The related hypothesis are the followings:

- H0: returns are normally distributed (if p-value>0.05)
- H1: returns are not normally distributed (if p-value<0.05)

The Jarque-Bera test is a goodness-of-fit test of whether sample data have the skewness and kurtosis matching a normal distribution. The related hypothesis are the followings:

- H0: returns are normally distributed (if p-value>0.05)
- H1: returns are not normally distributed (if p-value<0.05)

The P-value tends to zero for both tests in the first, third and fourth period, but in the Jarque-Bera test P-value is equal to 11.65%. This means that I can reject the null hypothesis of normal distribution at 11.65% . Also during the second period the Shapiro-Wilk test, P-value is 20.83%, which means that I can reject the null hypothesis of normal distribution at 20.83%.

To conclude this analysis, we will check for autocorrelation. The Autocorrelation (also known as “serial correlation”) is defined as the cross-correlation of a time series with itself at different point in time (that is why we use the word “cross”). In other words, it is a similarity between observations as a function of the time lag between them. In order to analysed whether the Gold is affected by autocorrelation or not, we will use graphical tools that refers to autocorrelation function (ACF). The correlograms plots the value that the ACF assumes for each lag k considered. Partial Autocorrelation Function (PACF) measures also the lag standing between t and t+k.

The ACF function says if the current value depends consistently on previous values (the lags). From chart 12, we can see the lags do not have significant effect (within the bounds – cannot tell them from being zero). We see that the only value is the spike at lag 0, so the daily values are independent of each other if only trying to explain them with themselves.

We will perform a statistical test regarding the autocorrelation of a time series called Ljung-Box Test, to be more precise. The Ljung-Box test whether any group of autocorrelations of a time series are different from zero. It test the “overall” randomness based on a number of lags. The Ljung-Box test may be defined as:

- H0: The data are independently distributed (if p-value>0.05)
- H1: The data are not independently distributed: they exhibit serial correlation (if p-value<0.05)

The tests on our data, in figure 2, confirm what we have seen in the correlogram above: by not refusing the null hypothesis, we have proof that in both the analysed periods Gold is not affected by autocorrelation. In fact, P-values are always higher than 0.05 (or 5%).

In conclusions, after performing all the statistical analyses, first on a large sample of rare metals related to the most important commodities in the world, and then on the selected rare metals, which is Gold, we can say that this asset is not affected by autocorrelation. It also doesn’t have a log-normal distribution, with the exception of period two from 1^{st} April 2014 to 31^{st} March 2015.

Packages used in R software to perform our analysis:

*Quandl*; *dplyr; xts; lubridate; dygraphs; sde; dtw; yuima; corrplot; e1071; tseries.*

## Recent Comments