Relationship between Foreign Exchange and Commodity Volatilities using High-Frequency Data

Transcription

1 Relationship between Foreign Exchange and Commodity Volatilities using High-Frequency Data Derrick Hang Economics 201 FS, Spring 2010 Academic honesty pledge that the assignment is in compliance with the Duke Community Standard as expressed on pp. 5-7 of "Academic Integrity at Duke: A Guide for Teachers and Undergraduates "

2 1. Introduction Understanding the factors that drive asset volatility, or risk, is critical to the overall understanding of movements in the financial markets. Several financial applications, such as those in derivative pricing and portfolio management, are all dependent on the ability to accurately model and predict volatility. Recent advancements in the financial theory, driven by the application of high-frequency data, have produced elegant models that are simpler to implement and have better measurement and forecasting capabilities than their predecessors. The Heterogeneous Autoregressive (HAR) model proposed by Corsi (2003), for instance, performs a simple linear regression to obtain estimates for realized variance. In the financial literature, several empirical studies exist that use these models to describe the activity in the foreign exchange markets; however, relatively few apply this framework to analyze the relationship between the volatility in foreign exchange with the volatility in commodities. For the most part, the link between the foreign exchange and commodity markets is welldocumented. Cashin et al. (200) finds evidence of a long-term connection between real exchange rates and commodity prices for approximately one-third of the commodity-exporting countries. Exchange rates that pair the currency of a heavily commodity-exporting country to the currency of a heavily commodity-importing country are seen in the market to follow the price movements of the underlying commodity. Chabin (2009) rationalizes this and similar findings by suggesting increases in commodity prices be viewed as a terms of trade improvement that is equivalent to a transfer of wealth from commodity-importing to commodity-exporting countries. Turning the focus of currency-commodity analysis to volatility, Diebold and Yilman (2010) find indications of volatility spillover from the US commodity to the US foreign exchange markets. We posit that, in particular, commodity currencies, due to their strong relationship with the commodity markets, are also more likely to exhibit a link in their volatilities; more specifically, we test for whether the volatility in the commodity can be indicative of the volatility of its respective commodity currency. We implement our analysis using both a simple linear regression approach and a more complex Bayesian method to provide a check for consistency in our results. In both of the models, we discover some evidence of commodity volatility to be indicative of the volatilities strong commodity currencies in the first half of

3 The rest of this paper is structured as follows: section 2 provides theoretical background behind the volatility measures and regression models implemented in this analysis; section 3 introduces the dataset of foreign exchange and commodity prices; section discusses research methods; section 5 presents the empirical results obtained from the regressions, and section 6 offers conclusive remarks about the findings and suggestions for further research. 2. Background 2.1 Stochastic Model of Returns Implicit in our calculations for realized variance in section 2.2, we assume the general stochastic model for price evolution, which indicates that the logarithm of prices p(t) of an asset can be defined by the following differential equation: dp t = u t dt + σ t dw t + κ t dq t (2.1.1) Where u(t) is the time varying drift component, σ(t)dw(t) is the time-varying volatility component, W t represents a standard Brownian motion, and σ(t) is the volatility level. κ(t)dq(t) represents a jump component where κ(t) is the magnitude of the jump and q(t) is a counting process. However, it is generally assumed that jumps occur infrequently. 2.2 Asset Return Volatility Models The geometric returns r t,j used in our model is obtained by taking the first lagged difference of the logarithm of intraday prices p t,j, shown below. Applying this formula in a rolling fashion, we acquire a series of intraday returns for each day t. r t,j = p t,j p t,j 1 (2.2.1) To obtain the values for the daily realized variance for our initial regressions, we took the sum of the m squared intraday returns for each day t. This measure of realized variance (RV) asymptotically approaches the integrated variance plus the jump component as the sampling frequency approaches infinity. m t RV t = r t,j 2 σ s 2 ds + κ s 2 (2.2.2) j =1 t 1 t 1<s t 3

4 In addition, we also considered the realized absolute value (RAV) measurement of volatility, which is defined to be the sum of the absolute value of intraday returns for each day t. As the sampling frequency approaches infinity, RAV asymptotically approaches the integrated volatility. RAV t = π 2m m j =1 r t,j t t 1 σ s ds It should be noted that RV and RAV are defined in different units and, therefore, cannot be compared directly. 2.3 Regression Models In this paper, we are largely dependent upon two different approaches, the Heterogeneous Autoregressive model and the Bayesian Dynamic Linear model, to perform our analysis Heterogeneous Autoregressive Models The first approach utilizes the Heterogeneous Autoregressive (HAR) model developed in Corsi (2003) for predicting realized variance. The HAR-RV model is a simple linear regression that is able to capture the persistence or time trends in an underlying time series by including its average lagged time series in the regression. Consider the following equation: RV t,t+ = 1 t+ k=t+1 RV k (2.3.1) Where RV t,t+h is the average RV over the given time span h. For the one-day ahead forecasting of RV t, we define a weekly regressor to be the average of the RVs of the past 5 days up to time t and a monthly regressor is the average of RVs of the past 22 days up to time t. We can now present the HAR-RV model to be RV t+1 = α + d RV t + w RV t 5,t + m RV t 22,t + ε t+1 (2.3.2)

5 Where RV t, RV t,t-5, RV t,t-22 represent lagged daily, weekly and monthly realized variance, RV t+1 is the one-day ahead forecasted realized variance, and d, w, m are the corresponding regression coefficient. Likewise, we also define a HAR-RAV model specified in the same fashion as the HAR-RV model, using RAV in place of RV. RAV t+1 = α + d RAV t + w RAV t 5,t + m RAV t 22,t + ε t+1 (2.3.3) Ghysels and Forberg (200) show that RAV regressors are more robust estimators of volatility than the RV sum of square measurement. However, note that the specific coefficient estimates obtained from the HAR-RAV model cannot be compared to those obtained from the HAR-RV model; in any case, it is still worthwhile to explore within-model significance to corroborate the overall findings Bayesian Models For our second approach, we chose to implement a univariate Bayesian dynamic linear model in place of the simple HAR models. The rationale behind this model choice is two-fold: the Bayesian framework provides a nice contrast to the previous analysis from which we can compare results to check for consistency; the Bayesian dynamic model allows for time-varying coefficient for the regressors, which provides another check for coefficient significance throughout the sample period. The full specifications and rationale behind the Bayesian dynamic linear model (DLM) can be found in West and Harrison (1997). Due to the complex nature of Bayesian modeling, we only include the descriptions of the major updating equations and assumptions in this paper. The general DLM model equations specified in our regression is as follows: y t = α t + 1,t y t 1 + 2,t x t 1 + v t were v t ~N 0, V t 1,t = 1,t 1 + ω 1,t were ω 1,t ~N 0, W 1,t (2.3.) 2,t = 2,t 1 + ω 2,t were ω 2,t ~N 0, W 2,t Where y t is the dependent variable, y t-1 is its first lagged term, x t-1 is an additional predictor, and v t is the observation error term which follows a normal distribution with mean 0, variance V t. 1,t 5

6 and 2,t are time-varying regression coefficients with evolution terms ω 1,t and ω 2,t which follow a normal distribution with zero mean and variance W 1,t and W 2,t respectively. The major model assumption is that the observational and the coefficient evolution terms can be modeled as a normal distribution, which violates the fundamental assumptions behind the calculations our volatility measures. In the case of realized variance, Anderson, Bollerslev, Diebold, and Labys (2000a) document that the distribution for realized variance is notably skewed, but move toward symmetry when we perform a logarithmic transformation on it. Thus, in our Bayesian analysis, we chose to model the logarithm of RV, and we note that the results from this model are not directly comparable to those obtained from the HAR models. However, as with the HAR-RAV analysis, within-model results from the Bayesian approach can still be used to corroborate the overall findings. The prior distribution for the coefficient i is defined as p i,t 1 D t 1 ~ T(m t 1, C t 1 ) (2.3.5) Where D t-1 is the data up to time t-1, m t-1 is the expected value of the coefficient, and C t-1 is the variance. Since we must specify an initial uninformative prior, there is a burn-in period for coefficient estimates before the model stabilizes. The posterior distribution for the coefficient i is defined as p i,t D t ~ T(m t, C t ) (2.3.6) Where the updating equations for m t and C t are m t = m t 1 + f W t Vt (y t m t 1 x t ) C t = (f W t V t ) y t m t-1 x t gives the expected portion of y not explained by x, and f represents a function. W t is the underlying variation in the coefficient term and V t is the observational error; their ratio represents a measure of how much the change in y t can be attributed to real movements of the coefficient rather than noise. Overall m t can be thought of simply as m t-1 plus movement in the underlying 6

7 level, and C t is determined by changes in the relationship between observational and evolution variances. 3. Data This dataset consists of 9 foreign exchange rates to the US dollar as well as Brent Crude Oil futures prices and Comex Gold future prices in US dollars, obtained from the vendor forextickdata.com, from January 2, 2009 to June 30, 2009, approximately 6 months worth of data. The data is high-frequency, providing 5-minute prices from 9:35AM to :00PM, excluding weekends. The currencies are listed as follows: AUDUSD, CHFUSD, EURUSD, GBPUSD, JPYUSD, NZDUSD, CADUSD, NOKUSD, ZARUSD. Currencies were chosen to be representative of the world markets although consideration was given to the expected magnitude of the effects of oil and gold on a given currency. Australia, Switzerland, New Zealand, and South Africa are large gold economies; Norway and Canada are large oil producers; and the United States is a large importer of both oil and gold. As such, we expect AUDUSD, CHFUSD, NZDUSD, and ZARUSD follow gold activity while CADUSD and NOKUSD follow oil activity, on the whole. Due to the sensitivity of the OLS analysis to outliers, we removed instances of extreme returns, or jumps, in order to lessen the influence of leverage points in our regressions. Not surprisingly, these extreme values correspond to unexpected macroeconomic announcements that occur within the time window. 3.1 Data Assumptions: Market microstructure noise Although the use of high-frequency data can greatly improve estimates of volatility, it also has the risk of capturing market microstructure noise, which represents the variation in the asset spot price from its fundamental value p t. Mathematically this is represented as p t = p t + e t (3.1.1) Where p t * is the spot price and e t is the deviation. This inclusion of microstructure noise in prices can distort the volatility measurements used in our analysis. Anderson, Bollerslev, Diebold, and Labys (2000b) advocate the use of a volatility signature plot, which presents a graph of the 7

8 average realized variance for each sampling frequency. The smallest sampling interval which displays a consistent value of average realized variance with those calculated with lower frequencies is considered optimal. Since the frequencies of our dataset can only be in intervals of 5 minutes, we could not construct a full signature plot; however, the average realized variance for the 5 minute frequency was relatively consistent to those of the 10 and 15 minute frequencies, which provide some support for our assumption that 5 minute price data is optimal.. Research Methodology.1 HAR regressions Consider the HAR-RV equation (2.3.2). To determine the possible influential effects of the RV of a commodity on the RV of a particular currency, we simply added the commodity RV at t-1 into the regression. RV curr :t+1 = α + d RV curr :t + w RV curr :t 5,t + m RV curr :t 2,t + comm RV comm :t + ε t+1 (.1.1) This regression is performed for each currency, for each commodity. Table 1 shows the oil RV coefficient estimate for each currency as well as its individual p-value, while table 3 presents the gold RV coefficient estimates and p-values. We then compare the statistical values obtained this model to those obtained the original HAR-RV without the commodity RV to assess significance. Table 2 and shows a side-by-side comparison of these models for the regressions in which the commodity RV coefficient was found to be individually significant. Now consider the HAR-RAV equation (2.3.3). The analysis for determining significance in commodity RAV follows the same methodology as described above; we just replace the RVs with their corresponding RAV components. The HAR- RAV equation with the inclusion of the t-1 commodity RAV is presented below. RAV curr :t+1 = α + d RAV curr :t + w RAV curr :t 5,t + m RAV curr :t 22,t + comm RV comm :t + ε t+1 (.1.2) Table 5 shows the oil RAV coefficient estimate for each currency as well as its individual p- value, while table 7 presents the gold RAV coefficient estimates and p-values. Again, we compare the statistical values obtained this model to those obtained the original HAR-RAV 8

9 without the commodity RAV to assess significance. Table 6 and 8 shows a side-by-side comparison of these models for the regressions in which the commodity RV coefficient was found to be individually significant..2 Bayesian DLM In the dynamic linear model, we perform a similar analysis, but allow the regression coefficients to be time-varying. The basic model is as follows: log (RV curr :t ) = α t + 1,t log(rv curr :t 1 ) + 2,t log(rv comm : t 1 ) + v t were v t ~N 0, V t 1,t = 1,t 1 + ω 1,t were ω 1,t ~N 0, W 1,t (.2.1) 2,t = 2,t 1 + ω 2,t were ω 2,t ~N 0, W 2,t Where RV t is the currency realized variance at time t, RV t-1 is the t-1 currency realized variance, RV comm:t-1 is the t-1 realized variance of the commodity, and v t is the observation error term which follows a normal distribution with mean 0, variance V t. 1,t and 2,t are the time-varying regression coefficients with evolution terms ω 1,t and ω 2,t following a normal distribution with zero mean and variance W 1,t and W 2,t respectively. Assumptions for the DLM are addressed in section Focusing on the posterior coefficient estimates and 95% credible intervals of the commodity RV regressor 2,t, we establish significance if its credible interval does not include zero for the entirety of the sampling period minus the month of January, which will be used for burn-ins. The Bayesian analysis of RAV was done using the same methodology described above, where the basic equations are analogous to equation (.2.1), but uses RAV instead of RV. log (RAV curr :t ) = α t + 1,t log(rav curr :t 1 ) + 2,t log(rav comm : t 1 ) + v t were v t ~N 0, V t 1,t = 1,t 1 + ω 1,t were ω 1,t ~N 0, W 1,t (.2.2) 2,t = 2,t 1 + ω 2,t were ω 2,t ~N 0, W 2,t.3 Additional Remarks Note that in our analysis, HAR-RV model is in variance terms, HAR-RAV model is in standard deviation terms, and the Bayesian DLM model is in logarithmic terms. Due to the theoretical backgrounds and assumptions of each model, as described above, results obtained from these 9

10 models are not directly comparable even if we were to perform transformations to a standardized term. Instead, the purpose of the analysis is to compare within model significance and see if this significance is corroborated among the different models. 5. Results 5.1 HAR-RV analysis Commodity coefficient estimates and their respective p-values for each currency HAR-RV regression are listed in table 1 and 3. From table 1, we find the coefficient for oil RV at time t-1 to only be significant in the regression for CADUSD RV with a p-value of Comparing the relative increase in fit by adding oil RV to the original HAR-RV for CADUSD, we see a relative large R 2 improvement of (see table 2). From table 3, we find the coefficient for gold RV at time t-1 to be significant in the regressions for AUDUSD and ZARUSD RVs with the p-values of and 0.031, respectively. Comparing the relative increase in fit by adding gold RV to the original HAR-RV model for these currencies, we see a modest R 2 improvement of for AUDUSD and 0.06 for ZARUSD (see table ). 5.2 HAR-RAV analysis Commodity coefficient estimates and their respective p-values for each currency HAR-RAV regression are listed in tables 5 and 7. From table 5, we find the coefficient for oil RAV at time t-1 to be significant in the regressions for CADUSD and NOKUSD RVs with p-values of and 0.08, respectively. Comparing the relative increase in fit by adding oil RV to the original HAR-RAV for CADUSD, we see a R 2 improvement of (see table 6). Similarly, the R 2 in the model for NOKUSD increased to Note that NOKUSD was not found to be significant at the 5% level in its corresponding HAR-RV regression. Looking at table 7, we find the coefficient for gold RAV at time t-1 to be significant in the AUDUSD, ZARUSD, and NZDUSD RAV regressions. Their corresponding p-values are 10

11 0.0183, 0.009, and 0.018, respectively. The overall R 2 improvements from adding in the gold RAV variable in the original HAR-RAV model are as follows: 0.0 for the AUDUSD RAV regression, for ZARUSD, and for NZDUSD (see table 8). In this case, notice that gold is significant at the 5% level in the HAR-RAV regression for NZDUSD and not in HAR- RV model for NZDUSD. 5.3 Bayesian Analysis Modeling the logarithm of realized variance with the dynamic linear model, we find similar results to the ones found from the HAR-RV analysis, namely that the oil coefficient was consistently significant from zero throughout the sampling period for the CADUSD RV regression, and that the gold coefficient was consistently significant for AUDUSD RV and ZARUSD RV. A plot of the time-varying posterior coefficient for gold in the AUDUSD RV regression is presented in figure 1, and a plot of the time-varying posterior coefficient for oil in the CADUSD RV regression is presented in figure 2. We do not include the month of January in the plots to account for model burn-in. Re-running the model using realized absolute variance, we also find matching results to the ones in obtained in the HAR-RAV model: the time-varying oil coefficient is significant through the sampling period for the CADUSD and NOKUSD regressions, and the time-varying gold coefficient is significant for the AUDUSD, ZARUSD, and NOKUSD regressions. The plot of the gold posterior RAV coefficient for the AUDUSD is presented in figure 3, and the oil posterior RAV for CADUSD is shown in figure. Note that the gold coefficient in the AUDUSD RAV regression is more or less stable around 0.3 while the oil coefficient in the CADUSD RAV regression starts out at 0.7 and gradually decline to stabilize around 0.2. The same pattern is seen in the complementary RV figures, 1 and 2. Conclusion Following previous work done on relationship between the commodity and foreign exchange markets, we extend the analysis to examine the link between the volatility of the two asset classes using the two high-frequency volatility measures, realized variance and realized absolute value. In particular, we are interested in the ability of a commodity s volatility to predict the volatility of a currency, especially those whose economies are largely dependent on that 11

12 particular commodity. We expect CADUSD and NOKUSD to follow oil more closely than other currencies and AUDUSD, CHFUSD, NZDUSD, and ZARUSD to be more related to gold than others (see section 3). After conducting comprehensive analysis in the HAR framework with both RV and RAV as well as performing corresponding analysis using a Bayesian DLM perspective, we find corroborating evidence among all of our models that oil volatility can be a useful predictor for CADUSD volatility and that gold volatility can be a useful predictor for the AUDUSD and ZARUSD volatilities. In our HAR and DLM models using realized absolute variance, we also find indications that oil volatility can be predictive of the volatility of NOKUSD and that gold volatility can be useful in NZDUSD volatility predictions although this is not seen in models using realized variance. Diebold and Yilman (2010) suggest that these instances of volatility spillovers can be attributed to general uncertainty caused by a global financial crisis and the onset of herd mentality. Indeed, they found that the overall spillover index in the markets rose to over thirty percent during the first half of 2009 as the effects of the financial crisis rippled through the world economies. Our findings are not inconsistent with these results. However, since our research only uses data from the first half of 2009, we can only conclude that there is evidence of significance in the commodity volatility regressors to predict volatility in foreign exchange for that time period. We are unable generalized our work to determine if this significance can be attributed to increased uncertainty in a financial crisis, an overall long-term relationship between these currencycommodity pairs, or a combination of the two effects. Further research can be done to explore this question by rerunning our analysis with a dataset consisting of more currency-pairs for a larger sampling period as well as incorporating the measurement of volatility spillover detailed in Diebold and Yilman (2009). 12

13 RV curr : t+1 = α + d RV curr :t + w RV curr :t 5,t + m RV curr :t 22,t + oil RV oil,t + ε t+1 AUDUSD CHFUSD EURUSD GBPUSD JPYUSD P-value NZDUSD CADUSD NOKUSD ZARUSD P-value Table 1: HAR-RV regressions: Oil RV coefficient and coefficient p-values for each currency regression α CADUSD (0.0001) CADUSD w/ RV oil (0.0001) (0.1012) (0.1016) (0.1027) (0.099) (0.0973) (0.098) 0.320** (0.103) P-value of F- Test R *indicates significance at 5% level, ** significant at 1% level Table 2: HAR-RV regressions: Model comparison (with standard deviations) between original HAR-RV model and the model with the oil RV for CADUSD 13

14 RV curr : t+1 = α + d RV curr :t + w RV curr :t 5,t + m RV curr :t 22,t + gold RV gold,t + ε t+1 AUDUSD CHFUSD EURUSD GBPUSD JPYUSD gold gold P-value NZDUSD CADUSD NOKUSD ZARUSD gold gold P-value Table 3: HAR-RV regressions: Gold RV coefficient and coefficient p-values for each currency regression AUDUSD AUDUSD w/ RV gold ZARUSD ZARUSD w/ RV gold α (0.0000) (0.0000) (0.0000) (0.0000) ** (0.091) 0.011** (0.0900) (0.0992) 0.19 (0.0988) (0.0927) (0.0912) (0.097) (0.0962) (0.1091) (0.1090) (0.0911) (0.0905) * (0.0303) * (0.0168) P-value of F- Test R *indicates significance at 5% level, ** significant at 1% level Table : HAR-RV regressions: Model comparison (with standard deviations) between original HAR-RV model and the model with the gold RV for AUDUSD and ZARUSD 1

15 RAV curr : t+1 = α + d RAV curr :t + w RAV curr :t 5,t + m RAV curr :t 22,t + oil RAV oil,t + ε t+1 AUDUSD CHFUSD EURUSD GBPUSD JPYUSD P-value NZDUSD CADUSD NOKUSD ZARUSD P-value Table 5: HAR-RAV regressions: Oil RAV coefficient and coefficient p-values for each currency regression CADUSD CADUSD w/ RAV oil NOKUSD NOKUSD w/ RAV oil α (0.0025) (0.002) (0.0012) (0.0012) (0.1003) (0.105) 0.323* (0.0991) * (0.102) (0.1013) (0.099) (0.0998) (0.100) (0.1037) (0.101) (0.088) (0.091) ** (0.0698) * (0.0329) P-value of F- Test R *indicates significance at 5% level, ** significant at 1% level Table 6: HAR-RAV regressions: Model comparison between original HAR-RAV model and the model with the oil RAV for CADUSD and NOKUSD 15

16 RAV curr : t+1 = α + d RAV curr :t + w RAV curr :t 5,t + m RAV curr :t 22,t + gold RAV gold,t + ε t+1 AUDUSD CHFUSD EURUSD GBPUSD JPYUSD gold gold P-value NZDUSD CADUSD NOKUSD ZARUSD gold gold P-value Table 7: HAR-RAV regressions: Gold RAV coefficient and coefficient p-values for each currency regression AUDUSD AUDUSD w/ RAV gold ZARUSD ZARUSD w/ RAV gold NZDUSD NZDUSD w/ RAV gold α (0.0010) (0.0010) (0.0010) (0.0009) (0.0013) (0.0012) ** (0.089) 0.257** (0.0892) ** (0.0983) ** (0.0961) ** (0.0938) ** (0.092) (0.093) (0.0921) (0.0961) (0.093) (0.093) (0.0932) (0.0867) (0.0901) (0.0878) (0.0898) (0.0902) (0.0983) * (0.095) ** (0.036) * (0.059) P-value of F-Test R *indicates significance at 5% level, ** significant at 1% level Table 8: HAR-RAV regressions: Model comparison between original HAR-RAV model and the model with the gold RAV for AUDUSD, ZARUSD, and NZDUSD 16

17 Bayesian Dynamic Linear Model Posterior Coefficient Plots with 95% Credible Intervals Figure 1 (above left): Graph of the time-varying RV gold coefficient when regressing for AUDUSD Figure 2 (above right): Graph of the time-varying RV oil coefficient when regressing for CADUSD Figure 3 (above left): Graph of the time-varying RAV gold coefficient when regressing for AUDUSD Figure (above right): Graph of the time-varying RAV oil coefficient when regressing for CADUSD 17

18 References Andersen, T., Bollerslev, T., Diebold, F.X. and Labys, P., "(Understanding, Optimizing, Using and Forecasting) Realized Volatility and Correlation," Published in revised form as "Great Realizations," Risk, September 2000(a), Andersen T. G., Bollerslev T., Diebold F. X., and Labys P., Microstructure bias and volatility signature, Unpublished Manuscript. 2000b. Cashin, P.,Céspedes, L.F. and Sahay, R., Commodity currencies and the real exchange rate, Journal of Development Economics, 75 (1) (200), pp Chaban, M., Commodity currencies and equity flows. Journal of International Money and Finance. Volume 28, Issue 5, September 2009, Pages Corsi, F., A Simple Long Memory of Realized Volatility. Unpublished Manuscript, University of Logano, Diebold, F.X. and Yilmaz, K., Measuring Financial Asset Return and Volatility Spillovers, With Application to Global Equity Markets, Economic Journal, 119 (2009), Diebold, F.X. and Yilmaz, K., Better to Give than to Receive: Predictive Directional Measurement of Volatility Spillovers. International Journal of Forecasting, March 1, Forthcoming. Ghysels, E, Forsberg, L, Why Do Absolute Returns Predict Volatility So Well? Journal of Financial Econometrics, 200. West, M., Harrison, J., Bayesian Forecasting and Dynamic Linear Models, 2nd Ed. Springer Publications