1) The document analyzes the relationship between stock-commodity correlation and business cycles from 1991-2014 using regression analysis. It finds the stock-commodity correlation is positively related to periods of economic weakness, as evidenced by a positive relationship with default spread. 2) Regression models show stock-commodity correlation is serially correlated and has a negative relationship with default spread, indicating higher correlation during recessions. However, the effect of real GDP growth and inflation on correlation is unclear. 3) In conclusion, the findings are consistent with prior research that stock-commodity correlation increases during economic downturns, when firms adjust behaviors and investor pessimism rises.

1. Regression Analysis:
Stock-Commodity Correlation and
Business Cycle
JIALU LI
Lehigh University
Instructed by Pr. David Muething
April 22, 2015
Abstract
Commodity market has been considered as good alternative in-
vestment market because of its low correlation with equity market
over long horizon. However, recent research conducted by [Bhard-
waj 2013] shows that this correlation tends to increase in recessions
because of macroeconomic factors such as firm conservative behaviors.
Following this idea, this paper focuses on studying the link between
stock-commodity correlation and general economy prosperity(GDP
growth, inflation and credit spread) during period (1991-2014). First,
the stock-commodity correlation is modeled based on the multi-stock
model, which is from Professor Vladimir Dobric’s Lecture Notes in
Financial Calculus. Then, a series of tests of heteroskedasticity, autocor-
relation are performed to test the data series. The regression result are
consistent with former researches: indeed the cross-market correlation
increases during depressions. Finally, this conclusion might serve as
important assumptions in risk management and portfolio investment.
Keywords: business cycle, stock-commodity correlation
1 Introduction
This article focuses on the relationship between business cycles and the stock-
commodity correlation. To check how the correlation between stock market
and commodity futures market can help us to examine the diversification
1

2. effectiveness across the stock market and the commodity market, especially
for recession periods.
Business cycle, which is mearsured by considering the real GDP growth
rate, pricing level, and etc., reflects the overall economic health condition
and fluctuations in a market economy. As widely accepted in the United
States, the National Bureau of Economic Research (NBER) defined the reces-
sion as "a significant decline in economic activity spread across the economy,
lasting more than a few mouths, normally visible in real GDP, real income,
employment, industrial production". The economic cycles also effect the
investment styles of portfolio managers. Portfolio managers strive to diver-
sify their risks by investing in multiple asset classes especially in economic
recessions.
Commodity futures, since its broad acceptance in 1990s, has been found
to be a desirable alternative investment class because of its low correlations
with the equity market. Commodity futures also has low intra-correlations
since different commodity categories are mainly effected by their own
supply-demand conditions, inventories and etc. Because of relatively low
stock-commodity correlation and intra-commodity correlation characteris-
tics, an increasing number of pension funds and sovereign wealth funds
are allocating more capital to commodities for the sake of diversification.
However, recent research conducted by [G. Bhardwaj and A. Dunsby 2013]
implies that the link between stock-commodity correlation and business
cycle is higher during periods of economic weakness and as well as the
average intra-commodity correlation. Therefore, my paper will take us to a
close look of this critical relationship.
2 Literature Review
Previous researches on business cycle and stock-commodity correlations has
found many interesting conclusions from differing perspectives. Overall low
stock-commodity correlation, equity-like returns and a positive correlation
with inflation are referred as the three main benefits of commodity investing;
see [G. Gorton and G. Rowenhorst. 2006]. [Kat, H., and R. Oomen. 2007] show
that the behavior of correlation differs by different commodity sectors (Agri-
cultural and Non-Argicultrual). [Chong, J., and J. Miffre. 2010] study the
correlation of 1981-2006 and find that it has fallen over time and is de-
creasing in equity volatility. [H. Zapata, J. Detre and T. Hnanbuchi. 2012]
examined historical commodity and stock market performance, and found
that a general zero or negative correlation between stocks and commodities
2

3. exists over the past.
Interestingly, recent research shows that during bad times of economy
this conclusion – zero stock-commodity correlation – does not necessar-
ily holds true. [G. Bhardwaj and A. Dunsby 2013] suggests that investors’
pessimism and firms’ conventional behaviors during depressions makes a
higher value of correlation ρ for stock and commodity market. This theory
can be used to explain the spikes of ρ in the early 1980s and in the late 2000s.
On the other hand, [D. Hendry. 1995] pointed out that studying business
cycles effects has been difficult since the dynamic feature, heterogenerous
and non-stationary time series data, and that no firm theoretical foundations.
Time series data in economies are heterogenerous, non-stationary, time
dependent, and interdependent.
Data series that were used in the above paper includes:
1) Real GDP Growth;
2) De f ault Spread;
3) CPI Index;
4) Bloomberg Commodity Index and the S&P − GSCI Index.
Therefore, these time series data will be a start point for my paper.
3 Data
To study the relationship of business cycles and the stock-commodity corre-
lations, two parts of data need to be addressed. Clearly,
a) one is the business cycle indicators;
b) the other is the cross-market correlations.
For the business cycle, three classical macroeconomic indicators are se-
lected: Real GDP Growth, Default spread, and CPI Index, to reflect the
general prosperity level and fluctuations of market economy.
For the stock-commodity correlations, I derived them by realized covari-
ance models with reference to [Andersen, T., T. Bollerslev, P. Christoffersen, and F. Diebold. 2012]
and the formula from Professor Vladimir Dobric’s lecture notes (MATH
468). The commodity indices I study are S&P-GSCI Index (SPGSCITR) and
the Bloomberg Commodity Index Total Return Index (BCOMTR). Mean-
while, S&P 500, are applied as the stock market indicator. Specific details
are enclosed in the Table 1. Theoretically, the more frequent data we use,
3

4. the correlation is more effective. Here I used the weekly prices to calculate
correlations.
Table 1: Data Source
3.1 Stock-Commodity Correlation Model
The primary empirical data - stock-commodity correlations - result from
the application of financial modeling technique in forecasting of second
moments. Applying linear models to calculate realized volatility and covari-
ance give us the realized correlation in this paper.
First of all, we can use past returns to estimate realized volatility. The
specific model that I used is presented as below:
u =
1
n
1
∆t
n
∑
i=1
yi
σ =
1
n − 1
1
∆t
n
∑
i=1
(yi∆t − µ∆t)2
where (1) µ is the estimator of the annualized realized expected log return;
(2) σ is the annualized realized expected volatility over n periods; (3) yi is
the log return in period i; (4) ∆t is the time interval for each period.
For covariance modeling, it is based on Multi-Stock Model with N-
Dimensional independent Brownian Motion:
dP(t) = M(t)dt + Ω(t)1/2
dW(t)
where M(t) and Ω(t)1/2 denote the N ×1 instantaneous drift vector and the
N × N positive definite square-root of the covariance matrix, respectively.
4

5. W(t) denotes a N-dimensional vector of independent Brownian motions.
Then realized intra-asset covariance will be:
RCov
i,j
t = Σk=t−1,t−1+h,...,t−hri(k, h)rj(k, h)
where ri(k, h) is the tick by tick return for asset i over the discrete interval
[k,k + h]. RCov
i,j
t is the realized intra-asset covariance for asset i,j.
Thus, with the realized volatility and covariance, therefore, we can drive
correlations from the formula:
ρi,j =
RCov
i,j
t
σt
i
σt
j
One thing worth mention is that the weighted average of stock-commodity
correlation (Corrt) will be used in regression tests of this paper.
Table 2: Data Summary
Note: G and I are 100 times the original data.
3.2 Data Analyses and Graphical Overview
Before stepping into regression results, let us take a look of the data graph-
ically. Figure 1 & 2 display correlations of stock-commodity correlations,
with default spread and inflation over period 1991-2014, respectively.
From the graphs, we can observe that the time series: stock-commodity
correlation is non-stationary; it wonders up and down overtime and a huge
spike appears around 2008-2014, overlapping with the 08 financial crisis.
More interestingly, the time of the correlation jumped to about 0.5 level
overlaps with the spikes in inflation and default spread.
5

6. Figure 1 Stock Commodity Correlations, and Default Spread
6

7. Figure 2 Stock Commodity Correlations, and Inflation
7

8. 3.2.1 Correlogram and autocorrelation
Figure 3: Correlogram of Stock-Commodity Correlation
Figure 3 displays the correlogram of overlapping annual stock-commodity
correlation; the graph shows that it is correlated with first 5 or 6 lags, which
means it is correlated with last five or six quarters’ correlations. This result
is consistent with our forecast that stock-commodity are serially correlated.
To further study the autocorrelation, an regression analysis of Corrt =
ρCorrt−1 + et (t-1 for previous year) is performed, and the correlogram of its
residual is presented in figure 4.
8

9. Figure 4: Correlogram for residuals from Lag Correlation Model
3.2.2 Heteroskedasticity
As we discussed in literature review part: the time series data include stock-
commodity correlation generally presents heteroskedastic characteristic.
Therefore, all regression result and analyses that we performed Later ((1)-
(4)) will use generalized least square estimates.
9

10. 4 Regression Model & Hypotheses
4.1 Regression Model
To test the positive relationship between stock-commodity correlation ρ
and recessions in business cycle, a series of regression model is applied,
formerly:
Model1 : Corrt = β0 + ρCorrt−1 + βGGt + et (1)
Model2 : Corrt = β0 + ρCorrt−1 + βI It + et (2)
Model3 : Corrt = β0 + ρCorrt−1 + βSSt + et (3)
Model4 : Corrt = β0 + ρCorrt−1 + βGGt + βI It + βSSt + et (4)
where β0 is just constants, Corrt is the stock-commodity correlation is
year t while Corrt−1 is the correlation for previous year; Gt is the real GDP
growth rate for year t; It is the CPI growth rate, which implies the inflation
degree for year t; and St is the default spread for year t.
4.2 Hypotheses
As we know that Default Spread are higher and GDP are lower while in
weak economic conditions. Therefore, we hypothesis the coefficient of real
GDP growth be negative (βG < 0), and the coefficient of Default spread
be positive(βS > 0). And since stock-commodity correlation is serially
correlated, ρs are expected be positive.
5 Empirical Results and Analysis
In this section I will present the results of regressing stock-commodity
correlation on lagged stock-commodity correlation measured over an annual
window, simultaneous GDP growth, the default spread, and inflation. The
STATA version is STATA 13.1.
5.1 Result Summary
10

11. Table 3a: Regression Model Result Summary (BCOMTR-S&P500)
Table 3b: Regression Model Result Summary (SPGSCITR-S&P500)
Table 3 presents the regression result for separate correlation data series
(Table 3a: SPGSCITR and Table 3b: BCOMTR). From the result we can see
the general idea that Stock-Commodity Correlation overall has a positive
relation with default spread, and a negative relation with real GDP growth.
To make it clear, I performed the same regression with the weight average
correlation data series. The result is enclosed in Table 4.
11

12. Table 4: Regression Model Result Summary (Weighted Average Corrt)
Note: 1.t-statistics in parentheses;
2. *p<0.05, **p<0.01, ***p<0.001;
3. Data of GDP Growth rate and Inflation timed 100 in Table 3.
From table 3, we can draw the following conclusions:
A) From all models, both the coefficient of previous year stock-commodity
correlation (ρ) and default spread (βS) are statistically and econometrically
significant and consistent with a weak economy. The significance of coef-
ficient of ρ, implies that the next years? stock-commodity correlation has
a high chance to perform in a similar manner with its last year. In other
words, if last year the stock-commodity correlation is high, then the next
year would possibly be high; similarly, if the stock-commodity correlation is
low in the first year, then the next year would likely to be low.
B) From result for model 1 and 4, for real GDP growth, the coefficient is
not significant even though the value of βG is meaningful from economic
standpoint.
C) For inflation, the coefficient βI is negative with a t-statistic -0.75, not
statistically significant. The reason can be in many folds. First, the inflation
is likely to be effected by US Federal Reserve. Because of belief in the power
of government adjusting business cycles, monetary supply are served as
the macroeconomic tools, which often do reverse adjustments. Besides, in
12

13. reality, inflation itself is not necessarily linked with bad economies. Increases
in consumption and not enough monetary supply will also leads to inflation.
Moderate inflation degree many times helps keeping healthy economy
growth. Indeed, hyperinflation reflects weakness in economy; However,
this situation is very rare and generally happens because of dysfunction of
central banks, which does not fit for US.
D) Overall, we can see that the explanatory power of model 4 has a
significant increase compared to model 1 and 2. And model 3 had the best
explanatory power with the highest R2 and lowest AIC and BC. Model
3 & 4 fit the correlation curve better than model 1 & 2, which means the
business cycle indicators do contain some information for stock-commodity
correlation, especially default spread.
5.2 Test of coefficient significance
5.2.1 Significance of Coefficient of Real GDP Growth Rate: βG
The null hypothesis H0 : βG < 0.
Alternative hypothesis H1 : βG ≥ 0.
The t statistic of βG, t = 0.46 while the p-value for the test is 0.602 >
0.05. We cannot reject the null hypothesis H0 : βG < 0 at the α = 0.025
level of significance. We can conclude that whether βG less than 0 is not
sure. Therefore, the effect of Real GDP Growth towards stock-commodity
correlation is not necessarily positive or negative.
5.2.2 Significance of Coefficient of Default Spread: βS
The null hypothesis H0 : βS > 0.
Alternative hypothesis H1 : βS ≤ 0.
The t test result of βS, t = 3.44 while the p-value for the test is 0.000 <
0.05. Thus we reject the null hypothesis H0 : βS > 0 at the α = 0.05 level of
significance. We can conclude that βS is less than 0, thus stock-commodity
correlation has a negative relationship with default credit spread, implying
a positive link between stock-commodity correlation with recessions.
6 Conclusions and Further Considerations
6.1 Conclusions
This paper examined the linkage between stock-commodity correlation and
business cycles. Our conclusion is not of many surprises. Initially, the result
13

14. in this study in generally persistent with former studies: the correlation
between stocks and commodities is higher during periods of economic
weaknesses, with the evidence of Default Spread. This is because firms
adjusting their business plans and investor risk aversion during recessions,
the whole commodity market and security market tends to behave more
similarly. The investors tends to treat both asset classes (stocks and commod-
ity futures) the same because of their psychological risk aversions. Besides,
while the effect of real GDP growth and spreads has a clear effect direction
on stock-commodity correlation, the effect of inflation on this correlation is
no clear or trending. The reason that the inflation and Real GDP Growth is
not significant can be in many folds, as discussed before. Furthermore, the
lagged correlation is a important predictor of future correlations.
The implications for risk management and portfolio management is
clear and can be helpful in real practice. While commodity futures has
been viewed as especially effective in providing diversification of both stock
and bond portfolios, we need to pay more attention for negative economic
signs in GDP growth, credit situation and consumptions, etc. From risk
management perspectives, managers need to alert for signals.
Put in another way, commodity futures can be a very desirable choice
for diversification during upward economy cycles because of its stronger
negative correlation over long horizon. Among GDP Growth, Inflation, and
Default Spread, Default Spread is the most useful and clear predictor for
stock-commodity correlation. When bad signs, such as default spread spike,
happen in the market, portfolio managers need to be careful while allocating
investments; risk managers might need to change some critical assumptions
in related risk models.
6.2 Further Considerations
A lot explorations and extensions on this base can be done to improve this
model:
a) First, for different categories, commodity exhibits different level of
correlation with equity market. The overall commodity market index in
this paper can definitely by replaced by one specific commodity category.
(industrial, precious metal, livestock, etc.).
b) Second, For the purpose of investment diversification, the correlation
between commodity market and other market (bond, currency futures or
other derivatives) are worthy to explore.
c) Additionally, the data in the realization of cross market correlation
model can be expanded by higher frequency data. Since the goal of this
14

15. model is to observe long term trends, more frequent short term data might
be used to test for short term trends.
d) Moreover, the data amount we use here is very limited. More data for
different area and countries can be used to conduct further researches.
15

16. References
Andersen, T., T. Bollerslev, P. Christoffersen, and F. Diebold. 2012.
Torben G. Andersen, Tim Bollerslev, Peter F. Christoffersen, and
Francis X. Diebold. (2012) Financial Risk Measurement for Financial Risk
Management. Handbook of the Economics of Finance. Amsterdam:
Elsevier B.V.
Chong, J., and J. Miffre. 2010. James Chong and Joëlle Miffre. (2010). Con-
ditional Correlation and Volatility in Commodity Futures and Traditional Asset
Markets. Journal of Alternative Investments 12, no.3 (winter): 61-75.
D. Hendry. 1995. David F. Hendry. (1995). Econometrics and Business Cycle
Empirics. The Economic Journal 105, no. 433 (November): 1622-1636.
G. Bhardwaj and A. Dunsby 2013. Geetesh Bhardwaj, Adam Dunsby.
(2013). The Business Cycle and the Correlation between Stocks and Com-
modities. Journal of Investment Consulting 14, No. 2. 14-25.
G. Gorton and G. Rowenhorst. 2006. Gary Gorton and K. Geert Rouwen-
horst. (2006). Facts and Fantasies about Commodity Futures. Financial
Analysts Journal 62, no.2 (March): 47-68.
H. Zapata, J. Detre and T. Hnanbuchi. 2012. Hector O. Zapata, Joshua D.
Detre, and Tatsuya Hanabuchi. (2012) Historical Performance of Commodity
and Stock Markets. Journal of Agricultural and Applied Economics 44,
no.3 (August):339-357.
Kat, H., and R. Oomen. 2007. Harry M. Kat, and Roel C.A. Oomen. (2007).
What Every Investor Should Know About Commodities Part II: Multivariate
Return Analysis. Journal of Investment Management 5, no. 3 (third
quarter): 1-25.
16

17. Appendix A. MATLAB Code for Correlation Model
%% Input
clear;clc;
InputSheet = ’Data.xlsx’;
[S, Text]=xlsread(InputSheet, ’Price’); %read index price
T=size(S,1);
N=T-1;
%% log Return
n = size(S,1)-1;
for j=1:3,
for i=1:n,
y(i,j)=log(S(i+1,j)/S(i,j));
end
end
%% mu estimator (overlapping annual data)
n =52;
dT =1/52;
for i=1:93,
mu(i,1:3)=(1/dT) *(1/n)*sum(y((13*(i-1)+1):(13*(i-1)+52), 1:3));
end;
%% sigma estimator (overlapping annual data)
for j=1:3,
for i=1:93,
y1= y((13*(i-1)+1):(13*(i-1)+52),j);
x=(y1-mu(i,j)*dT).^2;
sigma(i,j)=((1/dT)*(1/(n-1))*sum(x))^0.5;
end
end
%% realized covariance and correlation (overlapping annual data)
for i=1:93,
vol13(i,1)= 1/sigma(i,1)*(1/dT)*(1/n)* sum((y((13*(i-1)+1):(13*(i-1)+52), 1)
-mu(i,1)*dT).*(y((13*(i-1)+1):(13*(i-1)+52),3) -mu(i,3)*dT));
vol13(i,2)= (sigma(i,3)^2 -vol13(i,1)^2)^(0.5);
rho13(i,1)= vol13(i,1) /sigma(i,3);
vol23(i,1)= 1/sigma(i,1)*(1/dT)*(1/n)* sum((y((13*(i-1)+1):(13*(i-1)+52), 2)
-mu(i,2)*dT).*(y((13*(i-1)+1):(13*(i-1)+52),3) -mu(i,3)*dT));
vol23(i,2)= (sigma(i,3)^2 -vol23(i,1)^2)^(0.5);
17

18. rho23(i,1)= vol23(i,1)/sigma(i,3);
end
%%% Quaterly Data, Q1 1991 to Q4 2014
n2 =13;
dT =1/52;
for i=1:96,
mu_2(i,1:3)=(1/dT) *(1/n2)*sum(y((13*(i-1)+1):(13*(i-1)+13), 1:3));
end;
%% sigma estimator (Quaterly Data)
for j=1:3,
for i=1:96,
y1= y((13*(i-1)+1):(13*(i-1)+13),j);
x=(y1-mu_2(i,j)*dT).^2;
sigma_2(i,j)=((1/dT)*(1/(n2-1))*sum(x))^0.5;
end
end
%% realized covariance and correlation (Quaterly Data)
for i=1:96,
vol13_2(i,1)= 1/sigma_2(i,1)*(1/dT)*(1/n2)* sum((y((13*(i-1)+1):(13*(i-1)+13), 1)
-mu_2(i,1)*dT).*(y((13*(i-1)+1):(13*(i-1)+13),3) -mu_2(i,3)*dT));
vol13_2(i,2)= (sigma_2(i,3)^2 -vol13_2(i,1)^2)^(0.5);
rho13_2(i,1)= vol13_2(i,1) /sigma_2(i,3);
vol23_2(i,1)= 1/sigma_2(i,1)*(1/dT)*(1/n2)* sum((y((13*(i-1)+1):(13*(i-1)+13), 2)
-mu_2(i,2)*dT).*(y((13*(i-1)+1):(13*(i-1)+13),3) -mu_2(i,3)*dT));
vol23_2(i,2)= (sigma_2(i,3)^2 -vol23_2(i,1)^2)^(0.5);
rho23_2(i,1)= vol23_2(i,1)/sigma_2(i,3);
end
%% Output
Rho_1 = [rho13, rho23];
Rho_2 =[rho13_2, rho23_2];
xlswrite(’Data2.xlsx’, Rho_1 , ’Correlation(Overlapping Yearly)’ , ’A1’);
xlswrite(’Data2.xlsx’, Rho_2 , ’Correlation(Quaterly)’ , ’A1’);
Appendix B. STATA Command
1. GLS estimates
% model 1
reg rhoa L4. rho1 g
predict e1, residuals
18

19. gen esq1 = e1^2
reg esq1 L4.rhoa g
predict v
glm rhoa L4.rhoa g [aweight =1/v]
reg rhoa L4.rhoa g [aweight =1/v]
% model 2
drop e1 esq v
reg rhoa L4. rhoa i
predict e1, residuals
gen esq1 = e1^2
reg esq1 L4.rho1 i
predict v
glm rhoa L4.rhoa i [aweight =1/v]
reg rhoa L4.rhoa i [aweight =1/v]
% model 3
drop e1 esq v
eg rhoa L4. rhoa ds
predict e1, residuals
gen esq1 = e1^2
reg esq1 L4.rho1 ds
predict v
glm rhoa L4.rhoa ds [aweight =1/v]
reg rhoa L4.rhoa ds [aweight =1/v]
% model 4
drop e1 esq v
eg rhoa L4. rhoa g i ds
predict e1, residuals
gen esq1 = e1^2
reg esq1 L4.rho1
predict v
glm rhoa L4.rhoa g i ds [aweight =1/v]
reg rhoa L4.rhoa g i ds [aweight =1/v]
19