Estimating Market Risk Premium

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Estimating Market Risk Premium


Risk Premium Time Series


Abstract[edit]

This paper evaluates the relationship between risk premium and the volatility of the stock market, with the aim to analyse one of the possibilities that may improve the predictability of the premium for stock market risk.

Numerous works suggest that future risk premium can be predicted using the average of historical stock market returns; however, this paper offers an alternative, more flexible approach applicable for shorter investment horizons.

The analysis revealed that there is a significant correlation between stock market volatility and risk premium only in the USA, while the only factor significantly influencing risk premium in Japan and Germany is the GDP growth.

Introduction[edit]

The Security Market Line

A contemporary approach (like of Dimson, March, and Staunton) to evaluating the real risk premium for stock market risk is done by utilising historical data to calculate the historical average stock market risk premium. This can be suggested to investors as a benchmark in order to avoid over or underinvestment. This approach is based on the assumption of a constant underlying risk in the market.

However, what Dimson, March, and Staunton following the contemporary approach, possibly managed to achieve, was to calculate 100 year average risk premium for average risk during 100 years. Statistically it might make sense; however, currently the traders may not be very interested in the demanded risk premium, for example, before the second World War or during times of the Bretton-Woods system.

In reality, the underlying stock market risk could have been different in different periods. In practice, the calculated average the historical risk premium might not be correct at any period except, when the risk level coincides with historical average value, or in very long periods of time where we may observe average value.

Numerous authors like Graham and Harvey show that time period of analysis is very important, as strong positive correlation between the risk and the reward is observable in the long run; while for the short time periods the evidence is mixed.

Dropping the assumption of constant underlying risk may significantly improve reliability of the risk premium evaluation in the short run. This requires appropriate choice of proxy for risk. According to definition, a risk premium is a premium demanded by investors for the uncertainty of returns. (Damodaran, 2002, 60). Intuitive proxy for the uncertainty is the standard deviation of stock market returns, which used in many different works like Li, Graham and Harvey, and others.

Derivation of a function R_{prem} = f(\delta m) where Rprem denotes the risk premium, and  \delta m denotes fluctuations of equity returns, might be a reliable instrument for the estimation of the cost of capital, for a specific period of time, helping investors to keep their calculations and predictions up-to-date, and not based on historical average, which has nothing in common with the previous day stock market closing. Therefore we would like to conduct a research, and try to answer the question: "What is the Link between the Volatility of Market Returns and the Risk Premium?"

The rest of the study is organized as follows. Section 3 outlines the need for the research and theoretical background behind it. Section 4 concerns with hypothesis development. Section 5 describes the data used. Section 6 goes into analysis 7 concludes the work.

Background[edit]

Theoretical Background[edit]

Two articles of Dimson, March, and Staunton "Risk and return in 20th and 21st centuries" and "Global evidence on equity risk premium" were the starting point of our interest in the origin of the risk premium. In both articles authors tried to calculate the historical risk premium, and based on those calculations suggested to investors the risk premium they should expect in order to avoid under and over investment in everyday calculations.

However, authors of articles on the historical risk premiums accept that their calculations may be unreliable as they require long enough time period, where 100 years is not sufficient, in addition to that, the risk premium could change over time due to the changes in underlying risk. Finally stock market outcomes may be influenced by other factors. Another article studied in the framework of Financial Economics course by Jagannatan and McGrattan "The CAPM debate" shows numerous problems related to the evaluation of the risk premium in relatively short time periods.

Moreover, in the sample of a short period of time, such factors like company size and book-to-market ratio were reported to have much more explanatory power on the risk premium comparatively to a 100-year long period.

There is a need for a reliable methodology for cost of capital calculation for companies in order to avoid over and under investing (valuing). Currently, investors in a search of a fair risk premium to demand in a certain period are free to turn to financial markets, but as it is argued by wide body of the research in behavioural finance, the financial markets may give misleading information.

Review of Literature[edit]

An extensive analysis of the literature revealed a number of researches done in the area of predicting excess return on risky investments.

The very first to offer the concept of dependence of the risk premium from the volatility of market were Eugene Fama and James MacBeth. The research of Zhenyu Wang confirmed that allowing risk premium to vary over time, instead of using 100-year average significantly improves the predictability of stock market returns. The research of Robert F. Whitelaw has empirically confirmed that there is a strong correlation between the expected return and the conditional volatility.

However, Glosten, Jagannathan, and Runkle made a research that provides evidence on negative relation between conditional expected monthly return and conditional variance of monthly return. Modified GARCH model is applied in the research. Work itself argues that there is no clear link between return and variance, along with this finding authors mention a number of works that found that there is a positive and a negative correlation, and works that find no link at all.

Most of the related works we found conclude that there is a positive correlation between risk premium and risk represented as volatility of a stock. All models used in these works include various variables and out of them we try to apply only those that were shown to be truly important and proved to be significant.

Hypothesis Development[edit]

The primary aim of this research is to analyze the link between the volatility of stock markets and the stock market risk premium. The analysis also includes control variables. Inclusion of control variables was motivated by the need to decrease omitted variable bias and test influence of the stock market volatility on risk premium along with other factors and compare the results. According to the findings we can derive conclusion how significant volatility is for explaining risk premiums, comparatively to other factors, which according to financial theory should influence risk premium. Based on the available financial literature we develop the following Hypothesis:

Hypothesis 1: Increase in standard deviation shall be compensated by the higher risk premium.
Hypothesis 2: Increase in inflation shall be compensated by the higher risk premium.
Hypothesis 3: On the background of the higher GDP growth investors are ready to tolerate lower risk premium.

Data[edit]

For the purpose of the research stock market indices are used for calculation of both the stock market volatility and the stock market return. Data was gathered for as long period as possible from Reuters database, International Securities Market Association, and International Federation of Stock Exchanges, stock exchanges, national statistical bureaus via Internet and by contacting them. Our sample in order to be statistically representable for a cross-country analysis consists of 42 biggest countries by stock market. Overall number of observations used in our research is 21 727. Total number of observations collected for our research is above 100 000.

Indices[edit]

Daily observations of the most inclusive market indices were used to calculate monthly standard deviations of stock market returns, as well as monthly realized returns converted into the realized risk premium by subtracting from monthly return a one-year US government bond in the respective period. All indices used in this research are expressed in USD in order to adjust data for currency risk. As we take all indices in USD, USD denominated bond shall be used as a risk-free rate.

The perfect approach for calculation of certain market’s realized return is to gather data for all stocks in the market and weight it by capitalization, adjusting stock returns on dividends afterwards, however due to understandable time and financial constraints this approach was inapplicable in this research. Therefore market indices were used as a representation of analyzed markets.

As some may argue, stock index might be a bad tool for a specific country stock market analysis, as it does not represent all companies in the market, moreover it represents mainly the best ones or blue chips, what is already a bias by itself.

However, company stock with higher liquidity and with larger capitalization, tend to be better able to react on market movements, as well as make faster reactions on different factors. Although, small listed companies react similarly on such factors as news, accounting reports, global changes (basic price movements), but the manner of reaction is less predictable, and largely depends on other factors for example bid/ask spread. This in turn makes the reaction essentially less noisy for the stocks, which are more efficient in terms of information included, than stocks, which are not included in the index. Thus index that gathers most of such companies is a feasible tool for characterization of the market, showing the dynamics and the market reactions.

The procedure of the stock market index selection was as follows – the first priority was to get data of the index including all the stocks in the respective markets, in case non were available – the second most inclusive was chosen. In the case, currently the most inclusive index was established only a few years ago, like BSE – 500 in India, the most inclusive index for 1995 was chosen. (In case of India it is BSE – 200 which is established in 1995 as at that time, the amount of listed companies was smaller than today). Some countries have small number of companies in country’s main index, Portugal, for instance, has only 20 companies in its major stock index, however that is attributable to the origin of the specific stock market itself, and is not under our influence.

GDP[edit]

Quarterly data for GDP for the last ten year period was gathered. GDP data was gathered in terms of home currency of the respective country. There is an argument for gathering GDP data in USD terms in order to account for changes in real terms, for example Argentina’s GDP was down by more than 60% in USD terms during the crisis in 2001, while drop in Peso terms was much more modest. However this approach was not applied as otherwise change in GDP of certain countries (like Germany or France) would represent more currency fluctuations, rather than economic processes in the country.

We used absolute numbers of GDP adjusted to inflation in a local currency for a specific quarter in order to calculate the growth rate, later extrapolated in to monthly data. As quarterly data is a representation of a three month progress, consequently we divided quarter percentage change by three and assigned it to a corresponding month assuming that the growth was constant during the quarter, as we had no alternative reliable mechanism. If it was first quarter, than January, February and March each stood for one third of first quarter real GDP growth rate of a respective year.

Inflation[edit]

Inflation is a highly important aspect of a country's economic development; as a result it was also included in our data sample. First, inflation rates influence interest rates in the economy. Second, inflation influences consumption. Third, it influences country attractiveness for the foreign investor. As a result it influences stock market, what can be obsrved by the market movements after quarterly inflation data going public.

For the purpose of our research we have gathered data on the seasonally adjusted Consumer Price Index (CPI) that was used in order to calculate the inflation rates.

Methodology[edit]

Step 1[edit]

For the analysis we use unbalanced panel data for 42 countries for last 10 years. Choice of the amount of countries was influenced by the minimum requirement of 40 countries for statistically significant cross country comparison. In order to decrease omitted variable bias numerous control variables, like inflation rate, which was suggested in Wayne E. Ferson and Campbell R. Harvey, were added to the analysis. GDP growth adjusted to the inflation was introduced to the model as it sets a framework for overall country development and stock market growth.

First we apply some instruments of multivariate statistics namely scatterplot, Andrews curves, and the parallel plot. Next we do a linear regression. All observations were pooled together for the purposes of the analysis. This approach implicitly assumes no or insignificant between and within panel data effects. General relationship analysis in the research is as follows:

(2.1.1)
 Rpr = f \times STD + a \times Inf + c \times GDP \,

Where Rpr is realized risk premium, STD is standard deviation, Inf is inflation, GDP is the GDP growth rate, and f, a, c are respective coefficients. However exact variables of the model are adjusted for specifics and possibilities of each of each of the applied models.

Multivariate Statistics[edit]

Using the following code we have generated the scatterplot:

Scatterplot of all Countries' Indexes
1  library("stats")
2  library("plot")
3  setsize(600, 400)
4  d=createdisplay(2,2)
5  x = readxls (..\all.dat)
6  std=x[ ,3]
7  inf=x[ ,1]
8  GDP=x[ ,2]
9  Rpr=x[ ,4]
10 y = x [ ,4|3|1|2]
11 names =" Rpr"~" std "~" inf"~" GDP"
12 plotscml (y, names) 

The scatterplot shows a slight negative relationship between the standard deviation of stock market returns and the risk premium. Huge dispersed cloud in the origin of the graphs is a harbinger of the little predictability of the data and the presence of significant amount of outliers.

Andrewsplot of all Countries' Indexes
Parallel coordinates plot of all Countries' Indexes

Via the “andrews” quantlet we have generated AndrewsPlot. The graph bellows shows the annual observations of USA (red) and Argentina (black). The plot provides a hint of the importance of the country (between) effects in the data. Though the relationships seem to be of the same direction, Argentina data is obviously more volatile.

Finally using the “plotpcp” command we have generated the parallel coordinates plot. The plot also shows the possible presence of the country specific effects (like higher dispertion of observations in Argentina, or the fact that stock market in USA was always less volatile than in the Argentina).



Regression

We have estimated the linear regression with the help of the following code:

1 x = readxls (..\All.dat)
2 std=x[ ,3]
3 inf=x[ ,1]
4 GDP=x[ ,2]
5 Rpr=x[ ,4]
6 y = x [ ,4|3|1|2]
7 names =" Rpr"~" std "~" inf"~" GDP"
8 plotscml (y, names )

ANOVA table

[ 2,] "A  N  O  V  A                   SS      df     MSS       F-test   P-value"
[ 3,] "_________________________________________________________________________"
[ 4,] "Regression                     0.667     3     0.222       1.905   0.1288"
[ 5,] "Residuals                     34.197   293     0.117"
[ 6,] "Total Variation               34.864   296     0.118"
[ 7,] ""
[ 8,] "Multiple R      = 0.13834"
[ 9,] "R^2             = 0.01914"
[10,] "Adjusted R^2    = 0.00909"
[11,] "Standard Error  = 0.34163"
[12,] ""
[13,] ""
[14,] "PARAMETERS         Beta         SE         StandB        t-test   P-value"
[15,] "________________________________________________________________________"
[16,] "b[ 0,]=          0.0172       0.0338       0.0000         0.508   0.6117"
[17,] "b[ 1,]=         -0.5700       0.8876      -0.0375        -0.642   0.5212"
[18,] "b[ 2,]=          0.2323       0.2028       0.0668         1.145   0.2530"
[19,] "b[ 3,]=          1.1623       0.5755       0.1170         2.020   0.0443"

shows that the regression is of no use. Regression coefficient a around zero. All parameter coefficients are insignificant (except GDP that is on the margin). Additionally the very regression is insignificant as we cannot reject the hypothesis that all coefficients are zero, due to P-value exceeding 0,05.

Step 2[edit]

The most feasible explanation of the first regression failure is the significance of between country effects. In other words it is difficult to find one regression that would explain the risk premium dynamics in so different countries like Turkey, Germany, Japan, Brazil etc.

As Xplore has no instruments for the in-depth panel data analysis our next step is to turn to the analysis of the specific countries. For that purpose we will look at USA and Japan as first and second markets in the world by capitalisation, as well as Germany.

In the country specific analysis we rely on the monthly returns and standard deviation of the NYSE and TSE all stock indexes as well as DAX. We also use seasonally adjusted monthly inflation and inflation adjusted GDP growth of all three countries. However first we run the descriptive statistics in order to check for some problems in the data. For that purpose we use the following code:

x = read ("…")
t=x[,3]
f1=x[,3|9]
z=aseq(0,120,0)
z1=t~z
Rpr=x[,9]
gr11=grash(Rpr)
gr11=setmask(gr11,"line","red")
gr12=grqqn(Rpr)
gr21=grbox(Rpr)
gr22=grhist(Rpr)
show(d,1,1,gr11, gr22)
show(d,1,2,gr12)
show(d,2,1,gr21)
f1=setmask(f1,"line")
z1=setmask(z1,"line","blue")
show(d,2,2,f1,z1)
descriptive(Rpr)

The reader may take a look on the example output for USA Risk premium.

USA Descriptive Statistics
[ 6,] " Mean           0.00679968" 
[ 7,] " Std.Error       0.0455598     Variance       0.00207569" 
[ 8,] " " 
[ 9,] " Minimum         -0.144067     Maximum          0.119292" 
[10,] " Range            0.263359" 
[11,] " " 
[12,] " Lowest cases                  Highest cases " 
[13,] "         84:     -0.144067              54:    0.0811326" 
[14,] "         95:     -0.132393              24:    0.0814432" 
[15,] "         78:     -0.107571              48:    0.0963542" 
[16,] "         97:     -0.106225              79:     0.112589" 
[17,] "         45:    -0.0784607              66:     0.119292" 
[18,] " " 
[19,] " Median         0.00799733" 
[20,] " 25% Quantile   -0.0197645     75% Quantile       0.0354" 
[21,] " " 
[22,] " Skewness        -0.545989     Kurtosis          4.05956" 
[23,] "                               Excess            1.05956" 
[24,] " " 
[25,] " Observations                    120" [26,] " Distinct observations           120"

The risk premium data in all three countries had an excess kurtosis an negative skewness. By observing graphs one would be able to notice volatility clustering and fat tails characteristical for GARCH processes. Descriptive statistics of other variables brought no useful information except the fact that though there are extreme observations and outliers (according to the box plot information) they do not exceed 5% of samples. Of course this conclusion can not be extrapolated to the standard deviation variable as it resembles more of a Chi distribution, while box plot assumes normality.

USA regression results show

[ 2,] "A  N  O  V  A                   SS      df     MSS       F-test   P-value" 
[ 3,] "_________________________________________________________________________" 
[ 4,] "Regression                     0.028     3     0.009       4.947   0.0029" 
[ 5,] "Residuals                      0.219   116     0.002" 
[ 6,] "Total Variation                0.247   119     0.002" 
[ 7,] "" 
[ 8,] "Multiple R      = 0.33680" 
[ 9,] "R^2             = 0.11343" 
[10,] "Adjusted R^2    = 0.09050" 
[11,] "Standard Error  = 0.04345" 
[12,] "" 
[13,] "" 
[14,] "PARAMETERS         Beta         SE         StandB        t-test   P-value" 
[15,] "________________________________________________________________________" 
[16,] "b[ 0,]=          0.0208       0.0132       0.0000         1.579   0.1171" 
[17,] "b[ 1,]=         -2.9390       0.9695      -0.2681        -3.032   0.0030" 
[18,] "b[ 2,]=          0.8886       1.6743       0.0467         0.531   0.5966" 
[19,] "b[ 3,]=          4.4456       2.3522       0.1661         1.890   0.0613"

That stock market volatility significantly influences stock market risk premiums, though with the opposite sign than expected. There is negative relationship between volatility and retuns in the USA at least in the period of 1995-2004. Other variables turned out to be insignificant. Though regression coefficient increased compared to the crossectional regression, it is still far from being satisfying.

For Japan the regression fails to explain a thing:

[ 2,] "A  N  O  V  A                   SS      df     MSS       F-test   P-value" 
[ 3,] "_________________________________________________________________________" 
[ 4,] "Regression                     0.003     3     0.001       0.193   0.9010" 
[ 5,] "Residuals                      0.602   116     0.005" 
[ 6,] "Total Variation                0.605   119     0.005" 
[ 7,] "" 
[ 8,] "Multiple R      = 0.07048" 
[ 9,] "R^2             = 0.00497" 
[10,] "Adjusted R^2    = -0.02077" 
[11,] "Standard Error  = 0.07205" 
[12,] "" 
[13,] "" 
[14,] "PARAMETERS         Beta         SE         StandB        t-test   P-value" 
[15,] "________________________________________________________________________" 
[16,] "b[ 0,]=          0.0025       0.0215       0.0000         0.114   0.9094" 
[17,] "b[ 1,]=         -0.7772       1.2185      -0.0604        -0.638   0.5248" 
[18,] "b[ 2,]=         -0.0595       1.8844      -0.0029        -0.032   0.9749" 
[19,] "b[ 3,]=          0.7988       2.8008       0.0270         0.285   0.7760"

Unfortunately the same disappointing result is also observed in the case of Germany:

[ 2,] "A  N  O  V  A                   SS      df     MSS       F-test   P-value" 
[ 3,] "_________________________________________________________________________" 
[ 4,] "Regression                     0.001     3     0.000       0.067   0.9774" 
[ 5,] "Residuals                      0.445   116     0.004" 
[ 6,] "Total Variation                0.446   119     0.004" 
[ 7,] "" 
[ 8,] "Multiple R      = 0.04153" 
[ 9,] "R^2             = 0.00172" 
[10,] "Adjusted R^2    = -0.02409" 
[11,] "Standard Error  = 0.06192" 
[12,] "" 
[13,] "" 
[14,] "PARAMETERS         Beta         SE         StandB        t-test   P-value" 
[15,] "________________________________________________________________________" 
[16,] "b[ 0,]=          0.0074       0.0062       0.0000         1.200   0.2327" 
[17,] "b[ 1,]=          0.0000       0.0000       0.0222         0.234   0.8152" 
[18,] "b[ 2,]=         -0.8612       2.0849      -0.0392        -0.413   0.6803" 
[19,] "b[ 3,]=         -0.0352       0.3573      -0.0092        -0.099   0.9216"

Step 3[edit]

Not so inspiring previous results may come from the fact that we do not take into account important considerations from Financial theory into our analysis. First of all if analysed markets are at least semi efficient (there is a wide body of research that shows that they are), prices shall incorporate all of the publicly available information. Therefore markets shall react on unexpected changes. The problem to solve than is how to measure unexpected changes? This requires some assumptions.

We assume that Investors base their expectations on the previous twelve-month period information. Magnitude of unexpected change was calculated by subtracting average value of the variable during the preceding 12 months (12 month moving average) from the newly announced. Authors realize that in general it is suggested to take average for the longer preceding period as investors analyzing past information look for a longer period than a year, however due to data constraints we had to limit the period for moving average calculation to one year.

The next important factor is information lag. Even though investors may observe stock markets behaviour in real time, information on GDP and inflation reaches them with some lag. In order to accommodate this effect we shall use one month lagged inflation and two months lagged GDP growth, assuming that this are the time laggs required for the information to reach the market.

Additionally, one would have to take into account the fact that investors have different time horizons. Simply put there are market participants who react to any change in the market, and there are those who step in only when the trend persists over some time. In order to take a look on the influence of different time horizons we will include both current and last months unexpected stock market deviations into the regression.

Finally we have to distinguish between realised risk premium and the risk premium demanded by investors. Unfortunately, we have no possibility to measure the premium for stock market risk demanded by investors, as this is personal information of each investor. However, change in the realized risk premium is a good indicator of change in the demanded risk premium. When, for instance, market becomes riskier market participants adjust expected risk premium and start to demand higher returns, consequently they begin to sell stocks as they do not satisfy their newly adjusted expected risk premium. As a result stock price goes down and the realized risk premium also goes down.

In other words, decrease in realized risk premium in response to unexpected increase in the stock market volatility is an indicator of the fact that market participants have increased demanded risk premium.

All above-mentioned information requires the revision of our hypothesis. In the new hypotheses we use "unexpected" changes in variables, as the major market moves take place when new information deviates from market expectations. The reason behind, is that current market expectations are included in the current price, when something unexpected happens - price adjusts to the new information.

Hypothesis 1a: Positive unexpected standard deviation decreases the same period realized risk premium. Volatility itself may be perceived by investors as a warning sign of increasing risk, as the higher are stock market fluctuations the less certain are future outcomes. So higher than expected volatility represents higher than expected risk and investors should adjust their demanded risk premium. Consequently this adjustment decreases realized risk premium.

Hypothesis 1b: Positive unexpected standard deviation of the previous period decreases current period realized risk premium.

Hypothesis 2: Positive unexpected inflation from the previous period decreases current period realized risk premium.

Hypothesis 3: Positive unexpected real GDP growth from the previous period increases current period realized risk premium. The new regression

We construct the new regression using the following code:

x = read ("..\quantlets\All.dat")
std=x[,5]
inf=x[,6]
GDP=x[,7]
Rpr=x[,9]
STD0=x[13:120,5]
STD1=x[12:119,5]
STD2=x[11:118,5]
STD3=x[10:117,5]
STD4=x[9:116,5]
STD5=x[8:115,5]
STD6=x[7:114,5]
STD7=x[6:113,5]
STD8=x[5:112,5]
STD9=x[4:111,5]
STD10=x[3:110,5]
STD11=x[2:109,5]
STD12=x[1:108,5]
mov=STD1~STD2~STD3~STD4~STD5~STD6~STD7~STD8~STD9~STD10~STD11~STD12
DSTD=STD0-mean(mov,2)
DSTD1=DSTD[2:107,1]
DSTD=DSTD[3:108,1]

GDP0=x[13:120,7]
…
Rprt=x[15:120,9]
res=linreg(Var,Rprt)

The regression results are: [ 2,] "A N O V A SS df MSS F-test P-value"

[ 3,] "_________________________________________________________________________"
[ 4,] "Regression                     0.049     4     0.012       6.758   0.0001"
[ 5,] "Residuals                      0.184   101     0.002"
[ 6,] "Total Variation                0.233   105     0.002"
[ 7,] ""
[ 8,] "Multiple R      = 0.45949"
[ 9,] "R^2             = 0.21113"
[10,] "Adjusted R^2    = 0.17989"
[11,] "Standard Error  = 0.04264"
[12,] ""
[13,] ""
[14,] "PARAMETERS         Beta         SE         StandB        t-test   P-value"
[15,] "________________________________________________________________________"
[16,] "b[ 0,]=          0.0048       0.0041       0.0000         1.152   0.2519"
[17,] "b[ 1,]=         -5.7207       1.2273      -0.4887        -4.661   0.0000"
[18,] "b[ 2,]=          4.7399       1.1927       0.4050         3.974   0.0001"
[19,] "b[ 3,]=         -1.6776       2.6696      -0.0576        -0.628   0.5311"
[20,] "b[ 4,]=          1.7078       1.6848       0.0906         1.014   0.3132"

GDP growth and Inflation have signs in accordance with the new hypothesis stated, but are insignificant. STD variable shows a curios dynamics. Increase in volatility tends to decrease realised risk premiums in the respective month, but in the next month risk premiums recover to the high extent. This may be an indicator of markets overshooting. In order to check for consistency we have run the same regression but only with either current or lagged standard deviation. Though the magnitude of the effects decreases, signs do not change, suggesting that the result is robust.

We also check the regression for normality of residuals, heteroscedasticity, multicollinearity and linearity.

The Jarque-Bera normality test is calculated as follows:


\mathit{JB} = \frac{n}{6} \left( S^2 + \frac{{\left( K-3 \right)}^2}{4} \right)\,

where n is number of observations, S skewness and K kurtosis. The null hypothesis is that tested distribution follows normal distribution. The alternative hypothesis is that tested distribution does not follow normal distribution. Therefore the specification is correct. JB statistics follows ChiSquared Distribution with two degrees of freedom.

The test shows that we can not reject the hypothesis of normality of regressions residuals.

Ramsey RESET tests a linear specification against a non-linear specification of the model. The null hypothesis is that the correct specification is linear. The alternative hypothesis is that the correct specification is non-linear.

The test conducted by constructing restricted and unrestricted regressions defined as:

\begin{bmatrix}Y_{R} & = &\beta_{1}+\beta_{2}\times X_{2}+\beta_{3}\times X_{3}\\
Y_{U} & = &\beta_{1}+\beta_{2}\times X_{2}+\beta_{3}\times X_{3}+\gamma \times Y^{2}...
\end{bmatrix}\,

Where X are independent variables Y is dependant, X^{2} is dependant variable squared and \beta are respective coefficients. Than both regressions are compared. An unrestricted regression having significantly more explanatory power is and rejects the linearity hypothesis.



F_{M,N-k-1} = {\frac{\left(R^2_{R}-R^2_{U}\right)}{M}}/{\frac{1-R^2_{U}}{\left(N-K\right)}}
\,

Where M is the number of restrictions, N is the number of observations, K is the number of parameters estimated in the unrestricted equation. F value of Ramsey test is 79 what is a clear indication of problems related to non-linearity.

The Variance inflation factor (VIF) test of multicollinearity, requires an estimation of regression coefficients for all independent variables regressed against other independent variables and dependent variable. VIF test is estimated via the ratio 1/(1-R2). According to the decision rule of thumb VIF shall not exceed 10. VIF factors of the model do stay well below 1,5 indicating the absence of the multicollinearity problems.

Finally, we test for heteroscedasticity, which implies time varying variance of errors. We resort for a rather simple rest of estimating errors, splitting them in two sub samples and comparing their variances. The ratio of variances shall follow F-distribution with n-1 degrees of freedom. (according to Gujarati book, which is in the works cited) The F-statistics of the test is 7,1% what is above 2,5% and below 97,5%, therefore we can assume the homoscedasticity.

An important question to ask is if USA results are generalisable. Regression analysis for Japan is not to the advantage of our latest specification:


[ 2,] "A  N  O  V  A                   SS      df     MSS       F-test   P-value"
[ 3,] "_________________________________________________________________________"
[ 4,] "Regression                     0.036     4     0.009       1.682   0.1601"
[ 5,] "Residuals                      0.536   101     0.005"
[ 6,] "Total Variation                0.572   105     0.005"
[ 7,] ""
[ 8,] "Multiple R      = 0.24988"
[ 9,] "R^2             = 0.06244"
[10,] "Adjusted R^2    = 0.02531"
[11,] "Standard Error  = 0.07285"
[12,] ""
[13,] ""
[14,] "PARAMETERS         Beta         SE         StandB        t-test   P-value"
[15,] "________________________________________________________________________"
[16,] "b[ 0,]=         -0.0102       0.0071       0.0000        -1.434   0.1547"
[17,] "b[ 1,]=         -0.0393       1.9719      -0.0019        -0.020   0.9842"
[18,] "b[ 2,]=          2.7207       1.9843       0.1344         1.371   0.1734"
[19,] "b[ 3,]=          0.0001       0.0023       0.0044         0.046   0.9635"
[20,] "b[ 4,]=          5.9879       2.8783       0.2011         2.080   0.0400"

None of the used variables proves to be significant for the analysis, with the exception of GDP. Pearsons R-squared is mediocre (only 6%), and F-statistics suggests that we can not reject the hypothesis of all our coefficients to be insignificant. Econometric tests are similar to the USA case with only problem of non-linearity revealed.

Germany regression analysis has very similar results to Japan.

[ 2,] "A  N  O  V  A                   SS      df     MSS       F-test   P-value"
[ 3,] "_________________________________________________________________________"
[ 4,] "Regression                     0.023     4     0.006       1.442   0.2256"
[ 5,] "Residuals                      0.398   101     0.004"
[ 6,] "Total Variation                0.421   105     0.004"
[ 7,] ""
[ 8,] "Multiple R      = 0.23244"
[ 9,] "R^2             = 0.05403"
[10,] "Adjusted R^2    = 0.01657"
[11,] "Standard Error  = 0.06278"
[12,] ""
[13,] ""
[14,] "PARAMETERS         Beta         SE         StandB        t-test   P-value"
[15,] "________________________________________________________________________"
[16,] "b[ 0,]=          0.0061       0.0061       0.0000         1.006   0.3169"
[17,] "b[ 1,]=         -2.3266       2.1568      -0.1067        -1.079   0.2833"
[18,] "b[ 2,]=         -2.8738       2.2116      -0.1283        -1.299   0.1967"
[19,] "b[ 3,]=         -0.0294       0.1078      -0.0267        -0.273   0.7856"
[20,] "b[ 4,]=         -0.7784       0.3959      -0.1922        -1.966   0.0520"

Only GDP is on the margin of significance. Regression coefficient and F-statistics are far from making us happy. Econometric tests show similar results to Japan and USA.

Conclusion[edit]

In this work we have tried to find the flexible approach to estimate stock market risk premium based on the volitility of stock market returns, GDP growth and Inflation. In the course of the analysis we came to realise the importance of country specific effects and investors' expectations and their impact on the reliability of our model.

The only country identified with the significant relationship between volatility and returns was USA. However, this relationship was able to explain only a minor part of the risk premium variation in time. Analysis could be further improved by taking into account GARCH characteristics of the risk premium variable and applying Multivariate GARCH, by changing assumptions of the investors expectations mechanisms, or by tackling non-linearity relationships revealed by econometric tests.

References[edit]

Comments[edit]

  • This is a XploRe lecture, not a lecture about risk, e.g. most of section "Review of Literature" is superfluous
  • Andrews plot: Order of variables?
  • x = readxls (..\All.dat) I doubt that this works in XploRe
  • Could have been the library panel helpful to you?
  • Nice reference list, but I'am wondering about the STATA books, not even referenced in the text, and the XploRe books are missing
  • A hint: a reference should be "Dimson, March, and Staunton (2000)" instead of just "Dimson, March, and Staunton"