Gold ‘Nougat’: A Quantitative Perspective

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Aurum (Latin for Gold) serves as an effective hedge against inflation as indicated astutely by the trajectory of real interest rates. The real interest rate is nominal interest rate after taking inflation into consideration. Irving Fisher’s equation for the same:

r ≈ i – π; where i is the nominal interest rate, π is the rate of inflation or πe for ‘inflation expectations’, r is the real interest rate

A graphical representation of the same can be seen below for US Treasury yield (the benchmark 10Y and long-tenure 30Y) and CPI numbers that serve as proxies for the nominal interest rate and inflation figures respectively.  Nominal interest rates for medium and long-term were kept artificially low as per ‘Operation Twist’ in H2 2011 in the US. Negative real interest rates move in tandem with the Gold rise as seen from the 10Y and 30Y rate comparison charts. However, to accurately prove the same and determine how well-behaved and integrated this relationship is, we need to cleanse the data of noise.

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Primarily the methodology  involves testing the naked time series for stationarity using the Augmented Dicky Fuller test. A result of non-stationarity viz., the presence of a unit-root (the characteristic equation of the series has a value of ≥ 1) indicates varying mean, variance and co-variance. This is disadvantageous as this property produces large negative deviations from the mean at the beginning of the time series and equally large positive deviations towards its end.

There are two popular models for non stationary series depending on the mean trend viz., deterministic (trend stationary) or stochastic (difference stationary). Stationarity can be induced for the former, by estimating and removing the trend from the data using the Hodrick-Prescott filtration methodology to help decompose the series into trend and cyclical components and by stripping out the cyclical component. Time series with a deterministic trend always revert to the trend in the long run (the effects of shocks are eventually eliminated).

Trend-stationary process, yt:

yt= μtt; where μt is a deterministic mean trend,
εt  is a stationary stochastic process with mean zero

The Box-Jenkins methodology of differencing the series D times can be employed for the stochastic stationary time series  that never recover from shocks to the system (the effects of shocks are permanent).

Difference-stationary process, yt:

D yt=μ+Ψ(L)εt; where D=(1-L)D is the D order of differencing,
Ψ(L)=(1+Ψ1L+Ψ2L2+⋯) is an infinite degree lag operator polynomial with absolutely summable coefficients and all roots lying outside the unit circle,
εt is an uncorrelated innovation process with mean zero

In other words, we mean at looking series, without trend, with constant variance over time and with no periodic fluctuations (seasonality) or autocorrelation.

E(Xt )=μ for all t

E(Xt2 )=σ2 for all t

cov(Xt,Xk )=cov(Xt+s,Xk+s )for all s

In other words, we want a series without trend, with constant variance over time and with no periodic fluctuations (seasonality) or autocorrelation. If all the variables are stationary, we carry out a Granger causality test using Vector Autoregression (VAR), else using the Vector Error Correction Model (VECM) if some of the variables are non-stationary. This test helps us determine a ‘likely relationship’ when there is no apparent relationship between two variables. Advanced tests viz. the Johansen test must be used to test for the number of cointegrating relationships using VECMs.

References:

Brooks, C.. Introductory Econometrics for Finance. 1st ed. Cambridge University Press, 2002

Hodrick, R., and Prescott, E.C., 1997. Postwar U.S. Business Cycles: An Empirical Investigation. Journal of Money, Credit, and Banking, 29 (1), 1–16.

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