Recently, I performed this same exercise on gold (NYSEARCA:GLD) prices, whereby I used four different structural multivariate regression models to determine what factors are most influential in the metal's pricing. I received a request to do the same for silver (NYSEARCA:SLV), so I will gladly do so.

Gold and silver trade in a very tight correlation. Since 1968, based on quarterly data, the two have a correlation of 0.91, in the standard range running from 1 (perfect correlation) to -1 (perfect inverse correlation). Outside of some supply/demand and ratio differentials that may make one more favorable to the other at certain points in time, the two trade similarly.

(*Source: macrotrends.net*)

Since 1971, gold and silver have effectively traded as "alternative currencies" after the US independently terminated the convertibility of the US dollar (NYSEARCA:UUP)(NYSEARCA:UDN) to gold, in a move since known as the "Nixon shock." This effectively brought an end to the Bretton Woods system. But it wasn't until late 1973, when the US began to enter recession, when gold and silver truly began to trade as their own currencies to hedge against the stagflationary forces impacting the US economy at the time.

As I mentioned in the previous post, gold and silver are unique commodities (essentially commodity/currency hybrids) in that they trade off broader macroeconomic themes, particularly with respect to the US economy. Conventional commodities that are consumable and otherwise have some high level of utility form markets largely around the factors specific to these items themselves; namely, supply, demand, and overall inventory levels. The macro fundamentals are important but ultimately less influential.

Accordingly, a set of pertinent independent variables would generally include the most fundamental defining characteristics of the US economy, by virtue of the fact that it's the largest and comprises about 25% of all global economic activity (responsible for $18.7 trillion of the world's ~$75.2 trillion in GDP). Given the modern-day connectedness of the global economy, using Eurozone based metrics might be fine as well, for example, but given the overall bulk of the US economy and how gold and silver has traded off it historically, US-based metrics are logically the most appropriate.

Therefore, silver prices will be regressed on US-based variables, including real GDP growth, inflation rate, current account, budget deficit/surplus, M2 money supply, and the effective federal funds rate set by the Federal Reserve.

**Data**

Data was collected on each variable back through Q1 1968, around the period at which silver prices began deriving some level of sensitivity to world macroeconomic factors. All of the data, with minor exception, were collected from the St. Louis Fed's economic data site, which provides one of the largest public economic databases. YCharts supplied US current account data through the present-day given the St. Louis Fed's site went through only Q4 2013.

Moreover, because data for this exercise was organized by quarterly intervals and fiscal deficit/surplus data is released annually, I linearized the data between data points to ensure no gaps (otherwise 75% of the observations would have been unused in the regressions). The fiscal deficit for 2016 hasn't yet been released, but I also linearized the intervals through the end of Q3 based on the estimate obtained here.

**Results**

I made four separate structural models based on distinct distribution parameters: OLS (normal distribution), t-distribution, quantile regression (estimates conditional median and other quantiles, rather than strictly the mean), and log-normal.

Depending on the model, 4-5 of the 6 independent variables came back as highly statistically significant (i.e., at the 1% level or better) in each regression: US inflation (based on the personal consumption expenditures price index, or PCEPI), the federal funds rate, federal deficit as a percentage of GDP, and the M2 money supply.

Current account as a percentage of GDP was significant at the 5% level (p-value = 0.011) in the log-normal model. It was significant at the 10% level in the OLS regression (p-value = 0.058) and in the t-distribution model (p-value = 0.082).

Whether to attribute importance to a variable that is just shy of the 5% significance level is really a judgment call. Many researchers use the 5% level as a hard cutoff, as it tells them that they have 95% certainty that their analysis is correct, in that the variable's influence on the dependent variable (i.e., the price of silver in this case) is genuine.

Real GDP was not statistically significant in any model. Accordingly, this variable - along with current account, when applicable - was tossed out when forming the final model for each parametric approach.

(*Source: author*)

The results of each final model, encompassing data from Q1 1968 through Q3 2016, are as follows:

__OLS__

__T-Distribution__

__Quantile Regression__

__Log-Normal__

(*Source: author*)

Piecing these results into a model is a matter of using the coefficients displayed in the diagrams and multiplying by the respective input and adding each term together (or essentially subtracting if a coefficient is negative) to form an output.

The PCEPI, effective federal funds rate, current account, and federal deficit/surplus are all measured as a percentage. The M2 money supply is measured in billions (e.g., $13.2 trillion would be entered in as "13,200"). If we were to test what each model might predict based on a rough approximation of our current economic scenario, I inputted the following: PCEPI (1.2%), effective federal funds rate (0.625%), current account as a percentage of GDP (-2.8%), federal deficit as a percentage of GDP (-3.0%), and M2 money supply ($13.2 trillion).

The estimates are shown below. I also estimated a 95% confidence interval (2.5%-97.5%) for what these regressions might predict at upper and lower bounds to get a sense of what kind of range they're providing:

(*Source: author*)

The log-normal model is very much an outlier in that it fits the natural logarithm of the dependent variable (silver) to a normal distribution as opposed to the actual numerical value itself. This mitigates the influence of outliers in the data, which may or may not be appropriate. This results in a 21% higher reading than the $21.30 average of the OLS/t-distribution/quantile ("OTQ") regression model set. On top of that, the extravagant range it provides renders it effectively of little to no use.

The OTQ model set provides a range between $19.59-$23.34 at the 50th percentile. In total, this group estimates silver's fair price to be somewhere between $16.54-$28.58 within a 95% confidence interval.

**General Findings**

With respect to general take-home points, higher US inflation, higher interest rates, a higher federal deficit (or lower federal surplus), and a higher money supply are bullish for silver. Unsurprisingly, this compares similarly to the factors influencing the price of gold.

Like with the exercise that pertained to gold, the results on the effect of the US current account were mixed. Either the variable was not statistically significant or less significant relative to the aforementioned four variables. When it was significant, its sign was not always straightforward.

Generally, we might expect a higher current account deficit (or lower current account surplus) to be bullish for silver, in that such is a net-negative for the US economy. A lower current account entails a drop in the country's trade balance or a reduction in other foreign income.

In the realm of foreign exchange, lower current account and fiscal budget readings typically go hand-in-hand and are negative for the US dollar, which holds a distinct negative correlation to gold (US dollar is in orange; gold in blue):

(*Source: macrotrends.net*)

Accordingly, we would generally believe that factors that have traditionally been bearish for the US dollar would be bullish for silver, and vice versa. But the current account's influence on the two metals had a more nuanced and ultimately unclear effect.

**Hypothetical Influence of Each Variable on Silver Prices**

Below I go through a variety of hypothetical scenarios as to how an increase in each variable - while controlling for all other variables - would influence the price of silver.

__PCEPI (inflation)__

A 100-bp increase in inflation (relative to where I have it set at 1.2%) is expected to increase silver prices by 4.9% in the OLS model, 2.5% in the t-distribution model, and 3.2% in the quantile regression model.

And although I'm inclined to disown the log-normal model entirely for its imprecision and general oddity, a 100-bp increase in the PCEPI predicted an increase in silver prices of 16.1%.

__Federal Funds Rate__

A 100-bp increase in the federal funds rate predicts an increase in silver prices by 10-13 bps. The log-normal predicts a 42-bp increase.

__Current Account__

No effect on the OTQ set. The log-normal predicts a price increase of 1-2 bps.

__Federal Deficit__

A 1% increase in the federal deficit increases silver prices by 9-13 bps in the OTQ set (43 bps in the log-normal).

__M2 Money Supply__

A 1% increase in the M2 money supply each worked to increase the price of silver by 1% in each OTQ model +/- 2-3 bps. The log-normal is once again quite different in that it predicts an increase of 2.4% increase.

Overall, we uncover the already well-known reality that silver's best use is as a hedge against inflation. Other factors are significant, but not quite as influential on the price of the metal.

**A Word on Using These Models For Making Investment Decisions**

I want to emphasize the fact that models of this nature should ** never** be naively used to form investment opinions. This should go without saying, but a few of the comments on the gold article of the same topic prompted me to emphasize this in a section of its own. In the markets that I trade, particularly commodities and currencies, I run a variety of quantitative models of varying methodologies to help form an opinion. Even then, I don't necessarily trade off of them. No single model should ever be blindly followed, as in many cases certain models are of absolutely zero value.

The structural modeling form discussed here is merely a complement - not a full-blown, indiscriminately trusted substitute - to all other forms of analysis. Even though the outputs of this particular form of modeling, using the inputs over the time range discussed, suggest that silver may be undervalued, this is not an implicit recommendation to buy the metal. It would the equivalent to deciding to short a stock on the sole basis of some particular form of DCF analysis spit out a reading that shares might be overvalued, which would be completely absurd.

The primary purpose of this exercise is determine the *relationship* between the predictor variables and the price of silver and the degree of this influence, as opposed to predicting the response variable. In many cases, depending on internal diagnostic readings specific to each model (e.g., R-squared among others), trying to perform the latter may not even be appropriate and consequently render very imprecise predictions. Accordingly, the output of each model isn't necessarily the primary point of focus.

Although this article covers the regression outputs from historical inputs dating back to 1968, I also run regression models based on distinct time periods (i.e., by decade and over recent history) to better understand how the market may have changed, or is changing, over time. I feel that such analysis of what drives the market is much more edifying than using this analysis technique to merely spit out an end value.

**Disclosure:** I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.

I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.