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This is a fun exercise to be conducted over a series of posts, as doing so illustrates several important economic principles and some elegant mathematics.…\" />\n\t<style type=\"text/css\">\n\t\t\t\t#title a { color: #444444 }\n\t\t<\/style>\n\n<script type='text/javascript' src='http://partner.googleadservices.com/gampad/google_service.js'>\n<\/script>\n<script type='text/javascript'>\nGS_googleAddAdSenseService(\"ca-pub-3443918307802676\");\nGS_googleEnableAllServices();\n<\/script>\n<script type='text/javascript'>\nGA_googleAddSlot(\"ca-pub-3443918307802676\", \"wpcom_below_post_adsafe\");\n<\/script>\n<script type='text/javascript'>\nGA_googleFetchAds();\n<\/script>\n<script type=\"text/javascript\"> \n\twindow.google_analytics_uacct = \"UA-52447-2\"; \n<\/script>\n\n<script type=\"text/javascript\">\n\tvar _gaq = _gaq || [];\n\t_gaq.push(['_setAccount', 'UA-52447-2']);\n\t_gaq.push(['_setDomainName', 'wordpress.com']);\n\t_gaq.push(['_initData']);\n\t_gaq.push(['_trackPageview']);\n\n\t(function() {\n\t\tvar ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true;\n\t\tga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js';\n\t\t(document.getElementsByTagName('head')[0] || document.getElementsByTagName('body')[0]).appendChild(ga);\n\t})();\n<\/script>\n<\/head>\n<body class=\"single single-post postid-10079 single-format-standard vigilance-light highlander-enabled highlander-light\">\n\t<div class=\"skip-content\"><a href=\"#content\">Skip to content<\/a><\/div>\n\t<div id=\"wrapper\">\n\t\t\t\t<div id=\"header\" class=\"clear\">\n\t\t\t<div id=\"title\"><a href=\"http://quantivity.wordpress.com\"><span>Quantivity<\/span><\/a><\/div>\t\t\t<div id=\"description\">\n\t\t\t\t<h2>Uncommon Returns through Quantitative and Algorithmic Trading<\/h2>\n\t\t\t<\/div>\n\t\t\t<div id=\"nav\">\n\t\t\t\t\t<ul class=\"menu\">\n\t\t<li class=\"page_item \"><a href=\"http://quantivity.wordpress.com\">Home<\/a><\/li>\n\t\t\t\t<li class=\"page_item page-item-2\"><a href=\"http://quantivity.wordpress.com/about/\">About<\/a><\/li>\n\t<\/ul>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t<div id=\"content\" class=\"pad\">\n\t\t\t\t\t\t\t\t\t<div id=\"post-10079\" class=\"post-10079 post type-post status-publish format-standard hentry category-uncategorized\">\n\t\t\t<div class=\"post-header\">\n\t\t\t\t<h1>Multi-Asset Market Regimes<\/h1>\n\t\t\t\t<div id=\"single-date\" class=\"date\">November 9, 2012<\/div>\n\t\t\t<\/div>\n\t\t\t<div class=\"meta clear\">\n\t\t\t\t<div class=\"tags\"><\/div>\n\t\t\t\t<div class=\"author\">\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t\t<div class=\"entry clear\">\n\t\t\t\t\t\t\t\t<p>An astute reader suggested reproducing the results from a recent article on regime analysis by Kritzman <em>et al.<\/em>, <a href=\"http://www.cfapubs.org/doi/abs/10.2469/faj.v68.n3.3\">Regime Shifts: Implications for Dynamic Strategies<\/a> in FAJ (May / June 2012). This is a fun exercise to be conducted over a series of posts, as doing so illustrates several important economic principles and some elegant mathematics.<\/p>\n<p>This post begins by identifying <em>macroeconomic market regimes arising from multi-asset economic activity<\/em>.<\/p>\n<p><span id=\"more-10079\"><\/span><\/p>\n<p>One big challenge in analyzing market regimes is <em>identification<\/em>, as they are not directly observable. As an unsupervised statistical learning problem, there is no verifiable “right answer”. Beyond identifying regimes, we also want to know the probability of being in a given regime at any given point in time. Finally, our economic activity is measured via time series.<\/p>\n<p>Fortunately, a standard ML technique exists which possesses these attributes: <em>hidden Markov model<\/em> (HMM). For readers unfamiliar with HMM, here is a brief summary on the theory relevant to our problem. See <cite><a href=\"http://books.google.com/books?id=LDDzvCsdVs8C\">Hidden Markov Models for Time Series<\/a><\/cite> by Zucchini and MacDonald (2009) for more details.<\/p>\n<hr />\nHMMs are useful because they estimate both unobserved regimes and corresponding probabilities of being in each regime at every point in time. The latter is termed “local decoding”, and expressed as the conditional state probabilities:<\/p>\n<p> <img src='http://s0.wp.com/latex.php?latex=%5Ctext%7BPr%7D%28C_t+%3D+i+%7C+%5Cbf%7BX%7D_T+%3D+%5Cbf%7Bx%7D_T%29+%3D+%5Cfrac%7B%5Ctext%7BPr%7D%28C_t+%3D+i%2C+%5Cbf%7BX%7D%5ET+%3D+%5Cbf%7Bx%7D%5ET%29%7D%7B%5Ctext%7BPr%7D%28%5Cbf%7BX%7D%5ET+%3D+%5Cbf%7Bx%7D%5ET%29%7D+&bg=ffffff&fg=333333&s=0' alt='\text{Pr}(C_t = i | \bf{X}_T = \bf{x}_T) = \frac{\text{Pr}(C_t = i, \bf{X}^T = \bf{x}^T)}{\text{Pr}(\bf{X}^T = \bf{x}^T)} ' title='\text{Pr}(C_t = i | \bf{X}_T = \bf{x}_T) = \frac{\text{Pr}(C_t = i, \bf{X}^T = \bf{x}^T)}{\text{Pr}(\bf{X}^T = \bf{x}^T)} ' class='latex' /><\/p>\n<p>In other words, the probability of being in regime <img src='http://s0.wp.com/latex.php?latex=i&bg=ffffff&fg=333333&s=0' alt='i' title='i' class='latex' /> at time <img src='http://s0.wp.com/latex.php?latex=t&bg=ffffff&fg=333333&s=0' alt='t' title='t' class='latex' /> given the observed data <img src='http://s0.wp.com/latex.php?latex=%5Cbf%7Bx%7D_T&bg=ffffff&fg=333333&s=0' alt='\bf{x}_T' title='\bf{x}_T' class='latex' />.<\/p>\n<p>Where <img src='http://s0.wp.com/latex.php?latex=%5Cbf%7BX%7D&bg=ffffff&fg=333333&s=0' alt='\bf{X}' title='\bf{X}' class='latex' /> is the observed time-series data and <img src='http://s0.wp.com/latex.php?latex=C&bg=ffffff&fg=333333&s=0' alt='C' title='C' class='latex' /> is an unobserved parameter process which conditional probability at time <img src='http://s0.wp.com/latex.php?latex=t&bg=ffffff&fg=333333&s=0' alt='t' title='t' class='latex' /> depends at most on the previous time <img src='http://s0.wp.com/latex.php?latex=%28t-1%29&bg=ffffff&fg=333333&s=0' alt='(t-1)' title='(t-1)' class='latex' />, also known as the Markov property:<\/p>\n<p> <img src='http://s0.wp.com/latex.php?latex=%5Ctext%7BPr%7D%28C_%7Bt%2B1%7D+%7C+C_t%2C+...%2C+C_1%29+%3D+%5Ctext%7BPr%7D%28C_%7Bt%2B1%7D+%7C+c_t%29+&bg=ffffff&fg=333333&s=0' alt='\text{Pr}(C_{t+1} | C_t, ..., C_1) = \text{Pr}(C_{t+1} | c_t) ' title='\text{Pr}(C_{t+1} | C_t, ..., C_1) = \text{Pr}(C_{t+1} | c_t) ' class='latex' /><\/p>\n<p>Thus, <img src='http://s0.wp.com/latex.php?latex=C&bg=ffffff&fg=333333&s=0' alt='C' title='C' class='latex' /> is an unobservable Markov chain; because of which the conditional probability at time <img src='http://s0.wp.com/latex.php?latex=t&bg=ffffff&fg=333333&s=0' alt='t' title='t' class='latex' /> nicely reduces to (see equation 5.6 in Zucchini, derived from 4.9):<\/p>\n<p> <img src='http://s0.wp.com/latex.php?latex=%5Ctext%7BPr%7D%28C_%7Bt%2B1%7D+%7C+C_t%2C+...%2C+C_1%29+%3D+%5Cfrac%7B%5Calpha_t%28i%29+%5Cbeta_t%28i%29%7D%7BL_T%7D+&bg=ffffff&fg=333333&s=0' alt='\text{Pr}(C_{t+1} | C_t, ..., C_1) = \frac{\alpha_t(i) \beta_t(i)}{L_T} ' title='\text{Pr}(C_{t+1} | C_t, ..., C_1) = \frac{\alpha_t(i) \beta_t(i)}{L_T} ' class='latex' /><\/p>\n<p>Where <img src='http://s0.wp.com/latex.php?latex=%5Calpha_t%28i%29&bg=ffffff&fg=333333&s=0' alt='\alpha_t(i)' title='\alpha_t(i)' class='latex' /> and <img src='http://s0.wp.com/latex.php?latex=%5Cbeta_t%28i%29&bg=ffffff&fg=333333&s=0' alt='\beta_t(i)' title='\beta_t(i)' class='latex' /> are forward and backward probabilities along the Markov chain <img src='http://s0.wp.com/latex.php?latex=C&bg=ffffff&fg=333333&s=0' alt='C' title='C' class='latex' /> at time <img src='http://s0.wp.com/latex.php?latex=t&bg=ffffff&fg=333333&s=0' alt='t' title='t' class='latex' /> (estimated via iterative maximum likelihood using <a href=\"http://en.wikipedia.org/wiki/Expectation%E2%80%93maximization_algorithm\">expectation maximization<\/a>), and <img src='http://s0.wp.com/latex.php?latex=L_T&bg=ffffff&fg=333333&s=0' alt='L_T' title='L_T' class='latex' /> is the likelihood at time <img src='http://s0.wp.com/latex.php?latex=T+&bg=ffffff&fg=333333&s=0' alt='T ' title='T ' class='latex' />.<\/p>\n<p>For those who prefer code, where <code>val<\/code> is observed time series <img src='http://s0.wp.com/latex.php?latex=%5Cbf%7BX%7D&bg=ffffff&fg=333333&s=0' alt='\bf{X}' title='\bf{X}' class='latex' /> and <code>numOfStates<\/code> is number of unobserved regimes for <img src='http://s0.wp.com/latex.php?latex=C&bg=ffffff&fg=333333&s=0' alt='C' title='C' class='latex' />:<\/p>\n<p><pre class=\"brush: r;\">\r\nhmmFit <- HMMFit(val, nStates=numOfStates)\r\nfb <- forwardBackward(hmmFit, val)\r\neu <- exp(fb$Alpha + fb$Beta - fb$LL)\r\n<\/pre><\/p>\n<p>Readers with ML background will recognize HMM as an elementary temporal graphical model (see ? 6.2 of <a href=\"http://books.google.com/books?id=7dzpHCHzNQ4C\">Koller and Friedman (2009)<\/a>).<\/p>\n<hr />\nWith this tiny bit of HMM theory, we can now formulate our regime analysis problem in economic terms and use a bit of R to solve it. <\/p>\n<p>Consider four measures of economic activity: US equities, real gross domestic product (GDP), inflation, and G10 currencies. Posit each measure of economic activity can be characterized at any point in time as being in either one of <em>two<\/em> states: stable with corresponding small downside volatility or contracting with corresponding high volatility. This broadly matches traditional wisdom, namely the macroeconomy is either acting “fairly normal” or is exceptional (either panic or exuberance).<\/p>\n<p>Worth noting is the obvious caveat that this bi-state model oversimplifies reality, in particular ignoring potential distinction between “growth” and “stagnancy” (of potential importance during 1970s and 2010s). Intuition is unclear <em>a priori<\/em> whether a bi- or tri-state model is preferable. A subsequent post will take up this model selection question, as it lacks an easy answer.<\/p>\n<p>Illustrating the conditional probabilities of this model graphically, where a value of 0 indicates 100% likelihood of being in the “normal” regime and value of 1 indicates 100% likelihood of being in “exceptional” regime:<\/p>\n<p><a href=\"http://quantivity.files.wordpress.com/2012/11/hmm-regime-state-probs.png\"><img src=\"http://quantivity.files.wordpress.com/2012/11/hmm-regime-state-probs.png?w=600&h=826\" alt=\"\" title=\"hmm-regime-state-probs\" width=\"600\" height=\"826\" class=\"aligncenter size-full wp-image-10161\" /><\/a><\/p>\n<p>These regimes match our economic recollection. Equities were normal through much of the 1990s and mid-2000, and in panic the remaining time. GDP growth was exceptionally strong in the mid-1990s, during recovery from dotcom bubble in early 2000s, and during recovery of mortgage bubble in late 2000s; growth during all other times was “normal”. Inflation was high during the late 1970s and flanking the mortgage bubble. Currencies were volatile throughout 1980s and 1990s and then again flanking the mortgage bubble.<\/p>\n<p>These regimes also illustrate just how unusual the mortgage bubble was in historical sense, as it is the only time in the past 30 years during which all four measures of macroeconomic activity were simultaneously in exceptional regime.<\/p>\n<hr />\nCode to replicate the above results, and quite a bit more to be discussed in subsequent posts.<\/p>\n<p>Note: US equity regime is estimated using daily returns from SPX, rather than equally-weighted basket of S&P 500 sector indices as in the original article. Doing so results in nearly identical equity regime conditional probabilities, hence SPX is chosen in recognition of Occam.<\/p>\n<p><pre class=\"brush: r;\">\r\nlibrary("RHmm")\r\nlibrary("TTR")\r\n\r\ndisplayKritzmanRegimes <- function()\r\n{\r\n # Display regimes from Kritzman et al. (2012), printing regime\r\n # statistics and plotting local decoding.\r\n \r\n equityRegime <- getEquityTurbulenceRegime()\r\n inflationRegime <- getInflationRegime()\r\n growthRegime <- getGrowthRegime()\r\n currencyTurbulenceRegime <- getCurrencyTurbulenceRegime()\r\n\r\n print(equityRegime)\r\n print(inflationRegime)\r\n print(growthRegime)\r\n print(currencyTurbulenceRegime)\r\n \r\n plotMarkovRegimes(equityRegime, "Equity (SPX)", plotDensity=F)\r\n plotMarkovRegimes(inflationRegime, "Inflation (CPIAUCNS)", plotDensity=F)\r\n plotMarkovRegimes(growthRegime, "Real GDP (GDPC1)", plotDensity=F)\r\n plotMarkovRegimes(currencyTurbulenceRegime, "G10 Currency Turbulence", \r\n plotDensity=F)\r\n \r\n plotLocalDecodings(list(equityRegimeTurbulence, growthRegime, inflationRegime, \r\n currencyTurbulenceRegime),\r\n list("US Equity (SPX)", "Real GDP (GDPC1)", \r\n "Inflation (CPIAUCNS)","G10 Currency Turbulence"),\r\n regimeNums=c(2,2,2,2))\r\n}\r\n\r\ngetEquityTurbulenceRegime <- function(startDate=as.Date("1977-12-01"), \r\n endDate=Sys.Date(), numOfStates=2)\r\n{\r\n # Estimate two-state markov (SPX-based) equity regime. In lieu of S&P 500\r\n # sector indices, use SPX instead.\r\n #\r\n # Args:\r\n # startDate: date which to begin panel for regime estimation\r\n # endDate: end which to end panel for regime estimation\r\n # numOfStates: number of hidden states in regime\r\n # \r\n # Returns: hmmFit from HMMFit(), suitable for display with plotMarkovRegime()\r\n \r\n spx <- dROC(getOhlcv(instrumentSymbol="^GSPC", startDate=startDate, \r\n endDate=endDate, quote=c("close")))\r\n spxTurb <- rollingTurbulence(spx, avgWidth=(250 * 10), \r\n covarWidth=(250 * 10))\r\n meanTurb <- apply.monthly(spxTurb, mean)\r\n estimateMarkovRegimes(meanTurb, numOfStates=numOfStates)\r\n}\r\n\r\ngetInflationRegime <- function(startDate=as.Date("1946-01-01"), endDate=Sys.Date(),\r\n numOfStates=2)\r\n{\r\n # Estimate two-state markov (CPI-based) inflation regime.\r\n #\r\n # Args:\r\n # startDate: date which to begin panel for regime estimation\r\n # endDate: end which to end panel for regime estimation\r\n # numOfStates: number of hidden states in regime\r\n # \r\n # Returns: hmmFit from HMMFit(), suitable for display with plotMarkovRegime()\r\n \r\n val <- 100 *dROC(getFREDData(symbol="CPIAUCNS", startDate=startDate, \r\n endDate=endDate))\r\n estimateMarkovRegimes(val, numOfStates=numOfStates)\r\n}\r\n\r\ngetGrowthRegime <- function(startDate=as.Date("1946-01-01"), \r\n endDate=as.Date("2012-12-31"), numOfStates=2)\r\n{\r\n # Estimate two-state markov (GDP-based) growth regime.\r\n #\r\n # Note: Growth regime appears to be bi-modal, and thus need to estimate\r\n # several times to get convergence on the regime reported by Kritzman.\r\n #\r\n # Args:\r\n # startDate: date which to begin panel for regime estimation\r\n # endDate: end which to end panel for regime estimation\r\n # numOfStates: number of hidden states in regime\r\n # \r\n # Returns: hmmFit from HMMFit(), suitable for display with plotMarkovRegime()\r\n \r\n val <- 100 * dROC(getFREDData(symbol="GDPC1", startDate=startDate, \r\n endDate=endDate))\r\n estimateMarkovRegimes(val, numOfStates=numOfStates)\r\n}\r\n\r\ngetCurrencyTurbulenceRegime <- function(startDate=as.Date("1971-01-01"), \r\n endDate=Sys.Date(), \r\n numOfStates=2)\r\n{\r\n # Estimate two-state markov (G10-based) currency turbulence regime.\r\n #\r\n # Args:\r\n # startDate: date which to begin panel for regime estimation\r\n # endDate: end which to end panel for regime estimation\r\n # numOfStates: number of hidden states in regime\r\n # \r\n # Returns: hmmFit from HMMFit(), suitable for display with plotMarkovRegime()\r\n \r\n g10rates <- getG10Currencies()\r\n avgg10rates <- xts(100 * rowMeans(dROC(g10rates), na.rm=T), \r\n order.by=last(index(g10rates), -1))\r\n turbG10rates <- rollingTurbulence(avgg10rates, avgWidth=(250 * 3), \r\n covarWidth=(250 * 3))\r\n meanTurbG10rates <- apply.monthly(turbG10rates, mean)\r\n \r\n estimateMarkovRegimes(meanTurbG10rates, numOfStates=numOfStates)\r\n}\r\n\r\nestimateMarkovRegimes <- function(val, numOfStates=2)\r\n{\r\n # Estimate n-state hidden markov model (HMM) for val.\r\n #\r\n # Args:\r\n # val: series\r\n # numOfStates: number of hidden states in HMM\r\n #\r\n # Returns: hmmFit from HMMFit(), suitable for display with plotMarkovRegime()\r\n \r\n hmmFit <- HMMFit(val, nStates=numOfStates)\r\n return (list(val=val, hmmFit=hmmFit))\r\n}\r\n\r\nplotLocalDecodings <- function(regimes, symbols, plotDateRange="1900::2012", \r\n regimeNums)\r\n{\r\n # Plot local decodings for a list of HMM regimes, optionally over a set\r\n # date range.\r\n #\r\n # Args:\r\n # regimes: list of regimes, as returned by estimateMarkovRegimes()\r\n # symbols: list of human-readable symbols for regimes\r\n # plotDateRange: option date over which to plot regime local decodings\r\n # regimeNums: index of HMM regime, into regimes, to plot\r\n \r\n oldpar <- par(mfrow=c(1,1))\r\n on.exit(par(oldpar))\r\n \r\n layout(c(1,2,3,4))\r\n \r\n # generate merge of local decodings\r\n localList <- lapply(c(1:length(regimes)), function(i) {\r\n \r\n regime <- regimes[[i]]\r\n fb <- forwardBackward(regime$hmmFit, regime$val)\r\n eu <- exp(fb$Alpha + fb$Beta - fb$LL)\r\n local <- xts(eu[,regimeNums[i]], index(regime$val))[plotDateRange]\r\n \r\n plota(local, type='l', plotX=T, col=drawColors[i], main=symbols[i])\r\n })\r\n}\r\n\r\nplotMarkovRegimes <- function(regime, symbol, plotDateRange="1900::2012",\r\n plotDensity=T, plotTimeSeries=T)\r\n{\r\n # Plot markov regimes from HMM: kernel densities and per-regime local decodings.\r\n #\r\n # Args:\r\n # hmmFit: fit for HMM, as generated by estimateMarkovRegimes()\r\n # symbol: human-readable description of series with markov regimes\r\n # plotDateRange: contiguous range of time which to plot\r\n \r\n val <- regime$val\r\n hmmFit <- regime$hmmFit\r\n \r\n # calculate local decoding\r\n fb <- forwardBackward(hmmFit, val)\r\n eu <- exp(fb$Alpha + fb$Beta - fb$LL)\r\n hmmMeans <- hmmFit$HMM$distribution$mean\r\n hmmSD <- sqrt(hmmFit$HMM$distribution$var)\r\n \r\n # plot kernel density with regime means\r\n oldpar <- par(mfrow=c(1,1))\r\n on.exit(par(oldpar))\r\n \r\n if (plotDensity)\r\n {\r\n plot(density(val), main=paste("Density with Regime Means:", symbol))\r\n abline(v=mean(val), lty=2)\r\n \r\n sapply(c(1:length(hmmMeans)), function(i) {\r\n abline(v=hmmMeans[i], lty=2, col=drawColors[(i+1)])\r\n curve(dnorm(x, hmmMeans[i], hmmSD[i]), add=T, lty=3,\r\n col=drawColors[(i+1)])\r\n })\r\n }\r\n\r\n # Plot time series of percent change and local decoding for each regime\r\n if (plotTimeSeries)\r\n {\r\n merged <- merge(val, eu)\r\n layout(c(1:(1+ncol(eu))))\r\n \r\n plota(merged[,1][plotDateRange], type='l', paste("Regime:", symbol), plotX=F)\r\n sapply(c(1:length(hmmMeans)), function(i) {\r\n abline(h=hmmMeans[i], lty=2, col=drawColors[(i+1)])\r\n })\r\n plota.legend("Percent Change:", drawColors[1], last(merged[,1]))\r\n \r\n sapply(c(1:ncol(eu)), function(i) {\r\n \r\n plota(xts(merged[,(i+1)], index(val))[plotDateRange], type='l', \r\n plotX=(i==(ncol(eu))),\r\n col=drawColors[(i+1)])\r\n plota.legend(paste0("Event Regime ", i, ":"), drawColors[(i+1)], \r\n last(merged[,(i+1)]))\r\n })\r\n }\r\n}\r\n\r\ndROC <- function(x, n=1)\r\n{\r\n # Return discrete rate-of-change (ROC) for a series, without padding\r\n ROC(x, n, type="discrete", na.pad=F)\r\n}\r\n<\/pre><\/p>\n\n<div class=\"wpadvert\" style=\"position: relative; width:300px; text-align: center; padding: 0; margin: 10px auto; overflow: hidden;\">\n<a style=\"position: absolute; text-align: left; display: block; font: 9px/1 sans-serif; text-decoration: underline;\" href=\"http://en.wordpress.com/about-these-ads/\" rel=\"nofollow\">About these ads<\/a>\n<script type=\"text/javascript\">\nGA_googleAddAttr(\"AdOpt\", \"1\");\nGA_googleAddAttr(\"Origin\", \"other\");\nGA_googleAddAttr(\"LangId\", \"1\");\nGA_googleAddAttr(\"Domain\", \"quantivity.wordpress.com\");\nGA_googleAddAttr(\"BlogId\", 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class='like needs-login sd-button' rel='nofollow'><span>Like<\/span><\/a><\/div><div class=\"wpl-count sd-like-count\">Be the first to like this.<\/div><\/div><\/div><\/div>\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<div class=\"post-footer\">\n\t\t\t\t<div class=\"categories\">from → <a href=\"http://quantivity.wordpress.com/category/uncategorized/\" title=\"View all posts in Uncategorized\" rel=\"category tag\">Uncategorized<\/a><\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"navigation post single clear\">\n\t\t\t<div class=\"alignleft\"><a href=\"http://quantivity.wordpress.com/2012/10/25/implicit-momentum-bias/\" rel=\"prev\">← Implicit Momentum Bias<\/a><\/div>\n\t\t\t<div class=\"alignright\" ><\/div>\n\t\t<\/div>\n\t\t\n<div id=\"comments\">\n\t<div class=\"comment-number\">\n\t\t<span>7 Comments<\/span>\n\t\t\t<a id=\"leavecomment\" href=\"#respond\" title=\"Leave One\">leave one →<\/a>\n\t\t<\/div>\n\t\t\t<ol class=\"commentlist\">\n\t\t\t\n\t\t\t<li class=\"comment even thread-even depth-1 highlander-comment\" id=\"comment-1477\" >\n\t\t\t<div class=\"c-grav\"><img alt='' src='http://2.gravatar.com/avatar/2218b278cf55d488f355814bbdff9ea1?s=60&d=identicon&r=G' class='avatar avatar-60' height='60' width='60' /><\/div>\n\t\t\t\t<div class=\"c-body\">\n\t\t\t\t\t<div class=\"c-head\">\n\t\t\t\t\t\tSam <span class=\"c-permalink\"><a href=\"http://quantivity.wordpress.com/2012/11/09/multi-asset-market-regimes/#comment-1477\">permalink<\/a><\/span>\t\t\t\t\t<\/div>\n\t\t\t\t\t<div class=\"c-date\">\n\t\t\t\t\t\tNovember 9, 2012 6:58 pm\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<p>Nice post, which package does the dROC, getOhlcv etc functions reside in?<\/p>\n\t\t\t\t\t\t\t\t\t\t<div class=\"reply\">\n\t\t\t\t\t\t<a class='comment-reply-link' href='/2012/11/09/multi-asset-market-regimes/?replytocom=1477#respond' onclick='return addComment.moveForm(\"comment-1477\", \"1477\", \"respond\", \"10079\")'>Reply<\/a>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n<ul class='children'>\n\n\t\t\t<li class=\"comment byuser comment-author-quantivity bypostauthor odd alt depth-2 highlander-comment\" id=\"comment-1478\" >\n\t\t\t<div class=\"c-grav\"><img alt='' src='http://1.gravatar.com/avatar/7f67f1b2910b55a3f40a983ffa830357?s=60&d=identicon&r=G' class='avatar avatar-60' height='60' width='60' /><\/div>\n\t\t\t\t<div class=\"c-body\">\n\t\t\t\t\t<div class=\"c-head\">\n\t\t\t\t\t\tquantivity <span class=\"c-permalink\"><a href=\"http://quantivity.wordpress.com/2012/11/09/multi-asset-market-regimes/#comment-1478\">permalink<\/a><\/span><span class=\"asterisk\">*<\/span>\t\t\t\t\t<\/div>\n\t\t\t\t\t<div class=\"c-date\">\n\t\t\t\t\t\tNovember 9, 2012 8:29 pm\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<p>@Sam: thanks for complement.<\/p>\n<p>Re functions: some are common utility functions, which are not currently packaged; others are private because they depend on access to a large private data warehouse, which cannot be replicated with standard quantfin libraries (<em>e.g.<\/em> <code>quantmod<\/code>, etc).<\/p>\n<p>If there is sufficient reader interest, I can make the utility functions available.<\/p>\n\t\t\t\t\t\t\t\t\t\t<div class=\"reply\">\n\t\t\t\t\t\t<a class='comment-reply-link' href='/2012/11/09/multi-asset-market-regimes/?replytocom=1478#respond' onclick='return addComment.moveForm(\"comment-1478\", \"1478\", \"respond\", \"10079\")'>Reply<\/a>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n<ul class='children'>\n\n\t\t\t<li class=\"comment even depth-3 highlander-comment\" id=\"comment-1479\" >\n\t\t\t<div class=\"c-grav\"><img alt='' src='http://1.gravatar.com/avatar/abce5aadb1ba4843e271ba601c3501ed?s=60&d=identicon&r=G' class='avatar avatar-60' height='60' width='60' /><\/div>\n\t\t\t\t<div class=\"c-body\">\n\t\t\t\t\t<div class=\"c-head\">\n\t\t\t\t\t\tEduardo <span class=\"c-permalink\"><a href=\"http://quantivity.wordpress.com/2012/11/09/multi-asset-market-regimes/#comment-1479\">permalink<\/a><\/span>\t\t\t\t\t<\/div>\n\t\t\t\t\t<div class=\"c-date\">\n\t\t\t\t\t\tNovember 9, 2012 11:53 pm\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<p>I am looking at the code and I guess dROC is the same as TTR::ROC using type=”discrete”? Then, I guess getOhlcv is the same as quantmod:getSymbols just that it looks in your DWH? If this is true, then the only function missing definition and source code is rollingTurbulence since I can not find it anywhere else in R universe. I would appreciate if you can share the code.<\/p>\n<p>Otherwise, I guess the simplest thing would be to use data from FRED downloading it using getSymbols and using src=”FRED” and the example would be reproducible completely…<\/p>\n<p>Any chance to share code asked for in the comment section of another great article from you <a href=\"http://quantivity.wordpress.com/2012/10/23/volume-clock-gaps-and-goog/\" rel=\"nofollow\">http://quantivity.wordpress.com/2012/10/23/volume-clock-gaps-and-goog/<\/a> ? Thank you.<\/p>\n\t\t\t\t\t\t\t\t\t\t<div class=\"reply\">\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n<\/li>\n\n\t\t\t<li class=\"comment byuser comment-author-quantivity bypostauthor odd alt depth-3 highlander-comment\" id=\"comment-1483\" >\n\t\t\t<div class=\"c-grav\"><img alt='' src='http://1.gravatar.com/avatar/7f67f1b2910b55a3f40a983ffa830357?s=60&d=identicon&r=G' class='avatar avatar-60' height='60' width='60' /><\/div>\n\t\t\t\t<div class=\"c-body\">\n\t\t\t\t\t<div class=\"c-head\">\n\t\t\t\t\t\tquantivity <span class=\"c-permalink\"><a href=\"http://quantivity.wordpress.com/2012/11/09/multi-asset-market-regimes/#comment-1483\">permalink<\/a><\/span><span class=\"asterisk\">*<\/span>\t\t\t\t\t<\/div>\n\t\t\t\t\t<div class=\"c-date\">\n\t\t\t\t\t\tNovember 10, 2012 8:58 am\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<p>@Eduardo: I added dROC to the post. I have a version of the data access functions functions which stub into <code>quantmod<\/code> equivalents (and vice versa, so these functions interoperate with OHLCV functions), which I can make available in a subsequent post depending on reader interest.<\/p>\n<p>Covering turbulence is on the agenda for subsequent post in this series, as this paper depends on it (not to mention Kritzman is an author on a preceding paper covering turbulence).<\/p>\n<p>Finally, I can share TAQ code in a post subsequent to this market regime series.<\/p>\n\t\t\t\t\t\t\t\t\t\t<div class=\"reply\">\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n\n\t\t\t<li class=\"comment even thread-odd thread-alt depth-1 highlander-comment\" id=\"comment-1493\" >\n\t\t\t<div class=\"c-grav\"><img alt='' src='http://1.gravatar.com/avatar/4515b06f9f1c037ccfad149c4c45a88e?s=60&d=identicon&r=G' class='avatar avatar-60' height='60' width='60' /><\/div>\n\t\t\t\t<div class=\"c-body\">\n\t\t\t\t\t<div class=\"c-head\">\n\t\t\t\t\t\tgj <span class=\"c-permalink\"><a href=\"http://quantivity.wordpress.com/2012/11/09/multi-asset-market-regimes/#comment-1493\">permalink<\/a><\/span>\t\t\t\t\t<\/div>\n\t\t\t\t\t<div class=\"c-date\">\n\t\t\t\t\t\tNovember 12, 2012 9:41 pm\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<p>Thanks for the hmmFit package pointer.<\/p>\n\t\t\t\t\t\t\t\t\t\t<div class=\"reply\">\n\t\t\t\t\t\t<a class='comment-reply-link' href='/2012/11/09/multi-asset-market-regimes/?replytocom=1493#respond' onclick='return addComment.moveForm(\"comment-1493\", \"1493\", \"respond\", \"10079\")'>Reply<\/a>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n<\/li>\n\n\t\t\t<li class=\"comment odd alt thread-even depth-1 highlander-comment\" id=\"comment-1505\" >\n\t\t\t<div class=\"c-grav\"><img alt='' src='http://0.gravatar.com/avatar/fde8b319ca36fe6605a65d21f818d2cd?s=60&d=identicon&r=G' class='avatar avatar-60' height='60' width='60' /><\/div>\n\t\t\t\t<div class=\"c-body\">\n\t\t\t\t\t<div class=\"c-head\">\n\t\t\t\t\t\tNiclas <span class=\"c-permalink\"><a href=\"http://quantivity.wordpress.com/2012/11/09/multi-asset-market-regimes/#comment-1505\">permalink<\/a><\/span>\t\t\t\t\t<\/div>\n\t\t\t\t\t<div class=\"c-date\">\n\t\t\t\t\t\tNovember 16, 2012 11:04 pm\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<p>Thank you for a great post. You mention you have a large private data warehouse. Which sources do you use? Any suggestions on how to store such quantity of data (H2, sql, csv, …). Any special R packages you recomend?<\/p>\n<p>Thanks, looking forward to your next post.<\/p>\n\t\t\t\t\t\t\t\t\t\t<div class=\"reply\">\n\t\t\t\t\t\t<a class='comment-reply-link' href='/2012/11/09/multi-asset-market-regimes/?replytocom=1505#respond' onclick='return addComment.moveForm(\"comment-1505\", \"1505\", \"respond\", \"10079\")'>Reply<\/a>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n<\/li>\n\n\t\t\t<li class=\"comment byuser comment-author-7ovevol even thread-odd thread-alt depth-1 highlander-comment\" id=\"comment-1521\" >\n\t\t\t<div class=\"c-grav\"><img alt='' src='http://0.gravatar.com/avatar/990b8e4bb954a47aed9f94395e6c04a7?s=60&d=identicon&r=G' class='avatar avatar-60' height='60' width='60' /><\/div>\n\t\t\t\t<div class=\"c-body\">\n\t\t\t\t\t<div class=\"c-head\">\n\t\t\t\t\t\t<a href='http://convolutedvol' rel='external nofollow' class='url'>Convoluted Volatility<\/a> <span class=\"c-permalink\"><a href=\"http://quantivity.wordpress.com/2012/11/09/multi-asset-market-regimes/#comment-1521\">permalink<\/a><\/span>\t\t\t\t\t<\/div>\n\t\t\t\t\t<div class=\"c-date\">\n\t\t\t\t\t\tNovember 24, 2012 6:11 pm\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<p>Reblogged this on <a href=\"http://convolutedvolatility.com/2012/11/25/multi-asset-market-regimes/\" rel=\"nofollow\">Convoluted Volatility – StatArb,VolArb; Macro.<\/a>.<\/p>\n\t\t\t\t\t\t\t\t\t\t<div class=\"reply\">\n\t\t\t\t\t\t<a class='comment-reply-link' href='/2012/11/09/multi-asset-market-regimes/?replytocom=1521#respond' onclick='return addComment.moveForm(\"comment-1521\", \"1521\", \"respond\", \"10079\")'>Reply<\/a>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n<\/li>\n\t\t<\/ol>\n\t\t<div class=\"navigation\">\n\t\t<div class=\"alignleft\"><\/div>\n\t\t<div class=\"alignright\"><\/div>\n\t<\/div>\n\t\t<\/div>\n\n\t\t\t\t\t\t\t\t<div id=\"respond\">\n\t\t\t\t<h3 id=\"reply-title\">Leave a Reply <small><a rel=\"nofollow\" 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This is a fun exercise to be conducted over a series of posts, as doing so illustrates several important economic principles and some elegant mathematics.…\" />\n\t<style type=\"text/css\">\n\t\t\t\t#title a { color: #444444 }\n\t\t<\/style>\n\n<script type='text/javascript' src='http://partner.googleadservices.com/gampad/google_service.js'>\n<\/script>\n<script type='text/javascript'>\nGS_googleAddAdSenseService(\"ca-pub-3443918307802676\");\nGS_googleEnableAllServices();\n<\/script>\n<script type='text/javascript'>\nGA_googleAddSlot(\"ca-pub-3443918307802676\", \"wpcom_below_post_adsafe\");\n<\/script>\n<script type='text/javascript'>\nGA_googleFetchAds();\n<\/script>\n<script type=\"text/javascript\"> \n\twindow.google_analytics_uacct = \"UA-52447-2\"; \n<\/script>\n\n<script type=\"text/javascript\">\n\tvar _gaq = _gaq || [];\n\t_gaq.push(['_setAccount', 'UA-52447-2']);\n\t_gaq.push(['_setDomainName', 'wordpress.com']);\n\t_gaq.push(['_initData']);\n\t_gaq.push(['_trackPageview']);\n\n\t(function() {\n\t\tvar ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true;\n\t\tga.src = ('https:' == document.location.protocol ? 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This is a fun exercise to be conducted over a series of posts, as doing so illustrates several important economic principles and some elegant mathematics.<\/p>\n<p>This post begins by identifying <em>macroeconomic market regimes arising from multi-asset economic activity<\/em>.<\/p>\n<p><span id=\"more-10079\"><\/span><\/p>\n<p>One big challenge in analyzing market regimes is <em>identification<\/em>, as they are not directly observable. As an unsupervised statistical learning problem, there is no verifiable “right answer”. Beyond identifying regimes, we also want to know the probability of being in a given regime at any given point in time. Finally, our economic activity is measured via time series.<\/p>\n<p>Fortunately, a standard ML technique exists which possesses these attributes: <em>hidden Markov model<\/em> (HMM). For readers unfamiliar with HMM, here is a brief summary on the theory relevant to our problem. See <cite><a href=\"http://books.google.com/books?id=LDDzvCsdVs8C\">Hidden Markov Models for Time Series<\/a><\/cite> by Zucchini and MacDonald (2009) for more details.<\/p>\n<hr />\nHMMs are useful because they estimate both unobserved regimes and corresponding probabilities of being in each regime at every point in time. The latter is termed “local decoding”, and expressed as the conditional state probabilities:<\/p>\n<p> <img src='http://s0.wp.com/latex.php?latex=%5Ctext%7BPr%7D%28C_t+%3D+i+%7C+%5Cbf%7BX%7D_T+%3D+%5Cbf%7Bx%7D_T%29+%3D+%5Cfrac%7B%5Ctext%7BPr%7D%28C_t+%3D+i%2C+%5Cbf%7BX%7D%5ET+%3D+%5Cbf%7Bx%7D%5ET%29%7D%7B%5Ctext%7BPr%7D%28%5Cbf%7BX%7D%5ET+%3D+%5Cbf%7Bx%7D%5ET%29%7D+&bg=ffffff&fg=333333&s=0' alt='\text{Pr}(C_t = i | \bf{X}_T = \bf{x}_T) = \frac{\text{Pr}(C_t = i, \bf{X}^T = \bf{x}^T)}{\text{Pr}(\bf{X}^T = \bf{x}^T)} ' title='\text{Pr}(C_t = i | \bf{X}_T = \bf{x}_T) = \frac{\text{Pr}(C_t = i, \bf{X}^T = \bf{x}^T)}{\text{Pr}(\bf{X}^T = \bf{x}^T)} ' class='latex' /><\/p>\n<p>In other words, the probability of being in regime <img src='http://s0.wp.com/latex.php?latex=i&bg=ffffff&fg=333333&s=0' alt='i' title='i' class='latex' /> at time <img src='http://s0.wp.com/latex.php?latex=t&bg=ffffff&fg=333333&s=0' alt='t' title='t' class='latex' /> given the observed data <img src='http://s0.wp.com/latex.php?latex=%5Cbf%7Bx%7D_T&bg=ffffff&fg=333333&s=0' alt='\bf{x}_T' title='\bf{x}_T' class='latex' />.<\/p>\n<p>Where <img src='http://s0.wp.com/latex.php?latex=%5Cbf%7BX%7D&bg=ffffff&fg=333333&s=0' alt='\bf{X}' title='\bf{X}' class='latex' /> is the observed time-series data and <img src='http://s0.wp.com/latex.php?latex=C&bg=ffffff&fg=333333&s=0' alt='C' title='C' class='latex' /> is an unobserved parameter process which conditional probability at time <img src='http://s0.wp.com/latex.php?latex=t&bg=ffffff&fg=333333&s=0' alt='t' title='t' class='latex' /> depends at most on the previous time <img src='http://s0.wp.com/latex.php?latex=%28t-1%29&bg=ffffff&fg=333333&s=0' alt='(t-1)' title='(t-1)' class='latex' />, also known as the Markov property:<\/p>\n<p> <img src='http://s0.wp.com/latex.php?latex=%5Ctext%7BPr%7D%28C_%7Bt%2B1%7D+%7C+C_t%2C+...%2C+C_1%29+%3D+%5Ctext%7BPr%7D%28C_%7Bt%2B1%7D+%7C+c_t%29+&bg=ffffff&fg=333333&s=0' alt='\text{Pr}(C_{t+1} | C_t, ..., C_1) = \text{Pr}(C_{t+1} | c_t) ' title='\text{Pr}(C_{t+1} | C_t, ..., C_1) = \text{Pr}(C_{t+1} | c_t) ' class='latex' /><\/p>\n<p>Thus, <img src='http://s0.wp.com/latex.php?latex=C&bg=ffffff&fg=333333&s=0' alt='C' title='C' class='latex' /> is an unobservable Markov chain; because of which the conditional probability at time <img src='http://s0.wp.com/latex.php?latex=t&bg=ffffff&fg=333333&s=0' alt='t' title='t' class='latex' /> nicely reduces to (see equation 5.6 in Zucchini, derived from 4.9):<\/p>\n<p> <img src='http://s0.wp.com/latex.php?latex=%5Ctext%7BPr%7D%28C_%7Bt%2B1%7D+%7C+C_t%2C+...%2C+C_1%29+%3D+%5Cfrac%7B%5Calpha_t%28i%29+%5Cbeta_t%28i%29%7D%7BL_T%7D+&bg=ffffff&fg=333333&s=0' alt='\text{Pr}(C_{t+1} | C_t, ..., C_1) = \frac{\alpha_t(i) \beta_t(i)}{L_T} ' title='\text{Pr}(C_{t+1} | C_t, ..., C_1) = \frac{\alpha_t(i) \beta_t(i)}{L_T} ' class='latex' /><\/p>\n<p>Where <img src='http://s0.wp.com/latex.php?latex=%5Calpha_t%28i%29&bg=ffffff&fg=333333&s=0' alt='\alpha_t(i)' title='\alpha_t(i)' class='latex' /> and <img src='http://s0.wp.com/latex.php?latex=%5Cbeta_t%28i%29&bg=ffffff&fg=333333&s=0' alt='\beta_t(i)' title='\beta_t(i)' class='latex' /> are forward and backward probabilities along the Markov chain <img src='http://s0.wp.com/latex.php?latex=C&bg=ffffff&fg=333333&s=0' alt='C' title='C' class='latex' /> at time <img src='http://s0.wp.com/latex.php?latex=t&bg=ffffff&fg=333333&s=0' alt='t' title='t' class='latex' /> (estimated via iterative maximum likelihood using <a href=\"http://en.wikipedia.org/wiki/Expectation%E2%80%93maximization_algorithm\">expectation maximization<\/a>), and <img src='http://s0.wp.com/latex.php?latex=L_T&bg=ffffff&fg=333333&s=0' alt='L_T' title='L_T' class='latex' /> is the likelihood at time <img src='http://s0.wp.com/latex.php?latex=T+&bg=ffffff&fg=333333&s=0' alt='T ' title='T ' class='latex' />.<\/p>\n<p>For those who prefer code, where <code>val<\/code> is observed time series <img src='http://s0.wp.com/latex.php?latex=%5Cbf%7BX%7D&bg=ffffff&fg=333333&s=0' alt='\bf{X}' title='\bf{X}' class='latex' /> and <code>numOfStates<\/code> is number of unobserved regimes for <img src='http://s0.wp.com/latex.php?latex=C&bg=ffffff&fg=333333&s=0' alt='C' title='C' class='latex' />:<\/p>\n<p><pre class=\"brush: r;\">\r\nhmmFit <- HMMFit(val, nStates=numOfStates)\r\nfb <- forwardBackward(hmmFit, val)\r\neu <- exp(fb$Alpha + fb$Beta - fb$LL)\r\n<\/pre><\/p>\n<p>Readers with ML background will recognize HMM as an elementary temporal graphical model (see ? 6.2 of <a href=\"http://books.google.com/books?id=7dzpHCHzNQ4C\">Koller and Friedman (2009)<\/a>).<\/p>\n<hr />\nWith this tiny bit of HMM theory, we can now formulate our regime analysis problem in economic terms and use a bit of R to solve it. <\/p>\n<p>Consider four measures of economic activity: US equities, real gross domestic product (GDP), inflation, and G10 currencies. Posit each measure of economic activity can be characterized at any point in time as being in either one of <em>two<\/em> states: stable with corresponding small downside volatility or contracting with corresponding high volatility. This broadly matches traditional wisdom, namely the macroeconomy is either acting “fairly normal” or is exceptional (either panic or exuberance).<\/p>\n<p>Worth noting is the obvious caveat that this bi-state model oversimplifies reality, in particular ignoring potential distinction between “growth” and “stagnancy” (of potential importance during 1970s and 2010s). Intuition is unclear <em>a priori<\/em> whether a bi- or tri-state model is preferable. A subsequent post will take up this model selection question, as it lacks an easy answer.<\/p>\n<p>Illustrating the conditional probabilities of this model graphically, where a value of 0 indicates 100% likelihood of being in the “normal” regime and value of 1 indicates 100% likelihood of being in “exceptional” regime:<\/p>\n<p><a href=\"http://quantivity.files.wordpress.com/2012/11/hmm-regime-state-probs.png\"><img src=\"http://quantivity.files.wordpress.com/2012/11/hmm-regime-state-probs.png?w=600&h=826\" alt=\"\" title=\"hmm-regime-state-probs\" width=\"600\" height=\"826\" class=\"aligncenter size-full wp-image-10161\" /><\/a><\/p>\n<p>These regimes match our economic recollection. Equities were normal through much of the 1990s and mid-2000, and in panic the remaining time. GDP growth was exceptionally strong in the mid-1990s, during recovery from dotcom bubble in early 2000s, and during recovery of mortgage bubble in late 2000s; growth during all other times was “normal”. Inflation was high during the late 1970s and flanking the mortgage bubble. Currencies were volatile throughout 1980s and 1990s and then again flanking the mortgage bubble.<\/p>\n<p>These regimes also illustrate just how unusual the mortgage bubble was in historical sense, as it is the only time in the past 30 years during which all four measures of macroeconomic activity were simultaneously in exceptional regime.<\/p>\n<hr />\nCode to replicate the above results, and quite a bit more to be discussed in subsequent posts.<\/p>\n<p>Note: US equity regime is estimated using daily returns from SPX, rather than equally-weighted basket of S&P 500 sector indices as in the original article. Doing so results in nearly identical equity regime conditional probabilities, hence SPX is chosen in recognition of Occam.<\/p>\n<p><pre class=\"brush: r;\">\r\nlibrary("RHmm")\r\nlibrary("TTR")\r\n\r\ndisplayKritzmanRegimes <- function()\r\n{\r\n # Display regimes from Kritzman et al. (2012), printing regime\r\n # statistics and plotting local decoding.\r\n \r\n equityRegime <- getEquityTurbulenceRegime()\r\n inflationRegime <- getInflationRegime()\r\n growthRegime <- getGrowthRegime()\r\n currencyTurbulenceRegime <- getCurrencyTurbulenceRegime()\r\n\r\n print(equityRegime)\r\n print(inflationRegime)\r\n print(growthRegime)\r\n print(currencyTurbulenceRegime)\r\n \r\n plotMarkovRegimes(equityRegime, "Equity (SPX)", plotDensity=F)\r\n plotMarkovRegimes(inflationRegime, "Inflation (CPIAUCNS)", plotDensity=F)\r\n plotMarkovRegimes(growthRegime, "Real GDP (GDPC1)", plotDensity=F)\r\n plotMarkovRegimes(currencyTurbulenceRegime, "G10 Currency Turbulence", \r\n plotDensity=F)\r\n \r\n plotLocalDecodings(list(equityRegimeTurbulence, growthRegime, inflationRegime, \r\n currencyTurbulenceRegime),\r\n list("US Equity (SPX)", "Real GDP (GDPC1)", \r\n "Inflation (CPIAUCNS)","G10 Currency Turbulence"),\r\n regimeNums=c(2,2,2,2))\r\n}\r\n\r\ngetEquityTurbulenceRegime <- function(startDate=as.Date("1977-12-01"), \r\n endDate=Sys.Date(), numOfStates=2)\r\n{\r\n # Estimate two-state markov (SPX-based) equity regime. In lieu of S&P 500\r\n # sector indices, use SPX instead.\r\n #\r\n # Args:\r\n # startDate: date which to begin panel for regime estimation\r\n # endDate: end which to end panel for regime estimation\r\n # numOfStates: number of hidden states in regime\r\n # \r\n # Returns: hmmFit from HMMFit(), suitable for display with plotMarkovRegime()\r\n \r\n spx <- dROC(getOhlcv(instrumentSymbol="^GSPC", startDate=startDate, \r\n endDate=endDate, quote=c("close")))\r\n spxTurb <- rollingTurbulence(spx, avgWidth=(250 * 10), \r\n covarWidth=(250 * 10))\r\n meanTurb <- apply.monthly(spxTurb, mean)\r\n estimateMarkovRegimes(meanTurb, numOfStates=numOfStates)\r\n}\r\n\r\ngetInflationRegime <- function(startDate=as.Date("1946-01-01"), endDate=Sys.Date(),\r\n numOfStates=2)\r\n{\r\n # Estimate two-state markov (CPI-based) inflation regime.\r\n #\r\n # Args:\r\n # startDate: date which to begin panel for regime estimation\r\n # endDate: end which to end panel for regime estimation\r\n # numOfStates: number of hidden states in regime\r\n # \r\n # Returns: hmmFit from HMMFit(), suitable for display with plotMarkovRegime()\r\n \r\n val <- 100 *dROC(getFREDData(symbol="CPIAUCNS", startDate=startDate, \r\n endDate=endDate))\r\n estimateMarkovRegimes(val, numOfStates=numOfStates)\r\n}\r\n\r\ngetGrowthRegime <- function(startDate=as.Date("1946-01-01"), \r\n endDate=as.Date("2012-12-31"), numOfStates=2)\r\n{\r\n # Estimate two-state markov (GDP-based) growth regime.\r\n #\r\n # Note: Growth regime appears to be bi-modal, and thus need to estimate\r\n # several times to get convergence on the regime reported by Kritzman.\r\n #\r\n # Args:\r\n # startDate: date which to begin panel for regime estimation\r\n # endDate: end which to end panel for regime estimation\r\n # numOfStates: number of hidden states in regime\r\n # \r\n # Returns: hmmFit from HMMFit(), suitable for display with plotMarkovRegime()\r\n \r\n val <- 100 * dROC(getFREDData(symbol="GDPC1", startDate=startDate, \r\n endDate=endDate))\r\n estimateMarkovRegimes(val, numOfStates=numOfStates)\r\n}\r\n\r\ngetCurrencyTurbulenceRegime <- function(startDate=as.Date("1971-01-01"), \r\n endDate=Sys.Date(), \r\n numOfStates=2)\r\n{\r\n # Estimate two-state markov (G10-based) currency turbulence regime.\r\n #\r\n # Args:\r\n # startDate: date which to begin panel for regime estimation\r\n # endDate: end which to end panel for regime estimation\r\n # numOfStates: number of hidden states in regime\r\n # \r\n # Returns: hmmFit from HMMFit(), suitable for display with plotMarkovRegime()\r\n \r\n g10rates <- getG10Currencies()\r\n avgg10rates <- xts(100 * rowMeans(dROC(g10rates), na.rm=T), \r\n order.by=last(index(g10rates), -1))\r\n turbG10rates <- rollingTurbulence(avgg10rates, avgWidth=(250 * 3), \r\n covarWidth=(250 * 3))\r\n meanTurbG10rates <- apply.monthly(turbG10rates, mean)\r\n \r\n estimateMarkovRegimes(meanTurbG10rates, numOfStates=numOfStates)\r\n}\r\n\r\nestimateMarkovRegimes <- function(val, numOfStates=2)\r\n{\r\n # Estimate n-state hidden markov model (HMM) for val.\r\n #\r\n # Args:\r\n # val: series\r\n # numOfStates: number of hidden states in HMM\r\n #\r\n # Returns: hmmFit from HMMFit(), suitable for display with plotMarkovRegime()\r\n \r\n hmmFit <- HMMFit(val, nStates=numOfStates)\r\n return (list(val=val, hmmFit=hmmFit))\r\n}\r\n\r\nplotLocalDecodings <- function(regimes, symbols, plotDateRange="1900::2012", \r\n regimeNums)\r\n{\r\n # Plot local decodings for a list of HMM regimes, optionally over a set\r\n # date range.\r\n #\r\n # Args:\r\n # regimes: list of regimes, as returned by estimateMarkovRegimes()\r\n # symbols: list of human-readable symbols for regimes\r\n # plotDateRange: option date over which to plot regime local decodings\r\n # regimeNums: index of HMM regime, into regimes, to plot\r\n \r\n oldpar <- par(mfrow=c(1,1))\r\n on.exit(par(oldpar))\r\n \r\n layout(c(1,2,3,4))\r\n \r\n # generate merge of local decodings\r\n localList <- lapply(c(1:length(regimes)), function(i) {\r\n \r\n regime <- regimes[[i]]\r\n fb <- forwardBackward(regime$hmmFit, regime$val)\r\n eu <- exp(fb$Alpha + fb$Beta - fb$LL)\r\n local <- xts(eu[,regimeNums[i]], index(regime$val))[plotDateRange]\r\n \r\n plota(local, type='l', plotX=T, col=drawColors[i], main=symbols[i])\r\n })\r\n}\r\n\r\nplotMarkovRegimes <- function(regime, symbol, plotDateRange="1900::2012",\r\n plotDensity=T, plotTimeSeries=T)\r\n{\r\n # Plot markov regimes from HMM: kernel densities and per-regime local decodings.\r\n #\r\n # Args:\r\n # hmmFit: fit for HMM, as generated by estimateMarkovRegimes()\r\n # symbol: human-readable description of series with markov regimes\r\n # plotDateRange: contiguous range of time which to plot\r\n \r\n val <- regime$val\r\n hmmFit <- regime$hmmFit\r\n \r\n # calculate local decoding\r\n fb <- forwardBackward(hmmFit, val)\r\n eu <- exp(fb$Alpha + fb$Beta - fb$LL)\r\n hmmMeans <- hmmFit$HMM$distribution$mean\r\n hmmSD <- sqrt(hmmFit$HMM$distribution$var)\r\n \r\n # plot kernel density with regime means\r\n oldpar <- par(mfrow=c(1,1))\r\n on.exit(par(oldpar))\r\n \r\n if (plotDensity)\r\n {\r\n plot(density(val), main=paste("Density with Regime Means:", symbol))\r\n abline(v=mean(val), lty=2)\r\n \r\n sapply(c(1:length(hmmMeans)), function(i) {\r\n abline(v=hmmMeans[i], lty=2, col=drawColors[(i+1)])\r\n curve(dnorm(x, hmmMeans[i], hmmSD[i]), add=T, lty=3,\r\n col=drawColors[(i+1)])\r\n })\r\n }\r\n\r\n # Plot time series of percent change and local decoding for each regime\r\n if (plotTimeSeries)\r\n {\r\n merged <- merge(val, eu)\r\n layout(c(1:(1+ncol(eu))))\r\n \r\n plota(merged[,1][plotDateRange], type='l', paste("Regime:", symbol), plotX=F)\r\n sapply(c(1:length(hmmMeans)), function(i) {\r\n abline(h=hmmMeans[i], lty=2, col=drawColors[(i+1)])\r\n })\r\n plota.legend("Percent Change:", drawColors[1], last(merged[,1]))\r\n \r\n sapply(c(1:ncol(eu)), function(i) {\r\n \r\n plota(xts(merged[,(i+1)], index(val))[plotDateRange], type='l', \r\n plotX=(i==(ncol(eu))),\r\n col=drawColors[(i+1)])\r\n plota.legend(paste0("Event Regime ", i, ":"), drawColors[(i+1)], \r\n last(merged[,(i+1)]))\r\n })\r\n }\r\n}\r\n\r\ndROC <- function(x, n=1)\r\n{\r\n # Return discrete rate-of-change (ROC) for a series, without padding\r\n ROC(x, n, type="discrete", na.pad=F)\r\n}\r\n<\/pre><\/p>\n\n<div class=\"wpadvert\" style=\"position: relative; width:300px; text-align: center; padding: 0; margin: 10px auto; overflow: hidden;\">\n<a style=\"position: absolute; text-align: left; display: block; font: 9px/1 sans-serif; text-decoration: underline;\" href=\"http://en.wordpress.com/about-these-ads/\" rel=\"nofollow\">About these ads<\/a>\n<script type=\"text/javascript\">\nGA_googleAddAttr(\"AdOpt\", \"1\");\nGA_googleAddAttr(\"Origin\", \"other\");\nGA_googleAddAttr(\"LangId\", \"1\");\nGA_googleAddAttr(\"Domain\", \"quantivity.wordpress.com\");\nGA_googleAddAttr(\"BlogId\", 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class='like needs-login sd-button' rel='nofollow'><span>Like<\/span><\/a><\/div><div class=\"wpl-count sd-like-count\">Be the first to like this.<\/div><\/div><\/div><\/div>\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<div class=\"post-footer\">\n\t\t\t\t<div class=\"categories\">from → <a href=\"http://quantivity.wordpress.com/category/uncategorized/\" title=\"View all posts in Uncategorized\" rel=\"category tag\">Uncategorized<\/a><\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"navigation post single clear\">\n\t\t\t<div class=\"alignleft\"><a href=\"http://quantivity.wordpress.com/2012/10/25/implicit-momentum-bias/\" rel=\"prev\">← Implicit Momentum Bias<\/a><\/div>\n\t\t\t<div class=\"alignright\" ><\/div>\n\t\t<\/div>\n\t\t\n<div id=\"comments\">\n\t<div class=\"comment-number\">\n\t\t<span>7 Comments<\/span>\n\t\t\t<a id=\"leavecomment\" href=\"#respond\" title=\"Leave One\">leave one →<\/a>\n\t\t<\/div>\n\t\t\t<ol class=\"commentlist\">\n\t\t\t\n\t\t\t<li class=\"comment even thread-even depth-1 highlander-comment\" id=\"comment-1477\" >\n\t\t\t<div class=\"c-grav\"><img alt='' src='http://2.gravatar.com/avatar/2218b278cf55d488f355814bbdff9ea1?s=60&d=identicon&r=G' class='avatar avatar-60' height='60' width='60' /><\/div>\n\t\t\t\t<div class=\"c-body\">\n\t\t\t\t\t<div class=\"c-head\">\n\t\t\t\t\t\tSam <span class=\"c-permalink\"><a href=\"http://quantivity.wordpress.com/2012/11/09/multi-asset-market-regimes/#comment-1477\">permalink<\/a><\/span>\t\t\t\t\t<\/div>\n\t\t\t\t\t<div class=\"c-date\">\n\t\t\t\t\t\tNovember 9, 2012 6:58 pm\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<p>Nice post, which package does the dROC, getOhlcv etc functions reside in?<\/p>\n\t\t\t\t\t\t\t\t\t\t<div class=\"reply\">\n\t\t\t\t\t\t<a class='comment-reply-link' href='/2012/11/09/multi-asset-market-regimes/?replytocom=1477#respond' onclick='return addComment.moveForm(\"comment-1477\", \"1477\", \"respond\", \"10079\")'>Reply<\/a>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n<ul class='children'>\n\n\t\t\t<li class=\"comment byuser comment-author-quantivity bypostauthor odd alt depth-2 highlander-comment\" id=\"comment-1478\" >\n\t\t\t<div class=\"c-grav\"><img alt='' src='http://1.gravatar.com/avatar/7f67f1b2910b55a3f40a983ffa830357?s=60&d=identicon&r=G' class='avatar avatar-60' height='60' width='60' /><\/div>\n\t\t\t\t<div class=\"c-body\">\n\t\t\t\t\t<div class=\"c-head\">\n\t\t\t\t\t\tquantivity <span class=\"c-permalink\"><a href=\"http://quantivity.wordpress.com/2012/11/09/multi-asset-market-regimes/#comment-1478\">permalink<\/a><\/span><span class=\"asterisk\">*<\/span>\t\t\t\t\t<\/div>\n\t\t\t\t\t<div class=\"c-date\">\n\t\t\t\t\t\tNovember 9, 2012 8:29 pm\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<p>@Sam: thanks for complement.<\/p>\n<p>Re functions: some are common utility functions, which are not currently packaged; others are private because they depend on access to a large private data warehouse, which cannot be replicated with standard quantfin libraries (<em>e.g.<\/em> <code>quantmod<\/code>, etc).<\/p>\n<p>If there is sufficient reader interest, I can make the utility functions available.<\/p>\n\t\t\t\t\t\t\t\t\t\t<div class=\"reply\">\n\t\t\t\t\t\t<a class='comment-reply-link' href='/2012/11/09/multi-asset-market-regimes/?replytocom=1478#respond' onclick='return addComment.moveForm(\"comment-1478\", \"1478\", \"respond\", \"10079\")'>Reply<\/a>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n<ul class='children'>\n\n\t\t\t<li class=\"comment even depth-3 highlander-comment\" id=\"comment-1479\" >\n\t\t\t<div class=\"c-grav\"><img alt='' src='http://1.gravatar.com/avatar/abce5aadb1ba4843e271ba601c3501ed?s=60&d=identicon&r=G' class='avatar avatar-60' height='60' width='60' /><\/div>\n\t\t\t\t<div class=\"c-body\">\n\t\t\t\t\t<div class=\"c-head\">\n\t\t\t\t\t\tEduardo <span class=\"c-permalink\"><a href=\"http://quantivity.wordpress.com/2012/11/09/multi-asset-market-regimes/#comment-1479\">permalink<\/a><\/span>\t\t\t\t\t<\/div>\n\t\t\t\t\t<div class=\"c-date\">\n\t\t\t\t\t\tNovember 9, 2012 11:53 pm\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<p>I am looking at the code and I guess dROC is the same as TTR::ROC using type=”discrete”? Then, I guess getOhlcv is the same as quantmod:getSymbols just that it looks in your DWH? If this is true, then the only function missing definition and source code is rollingTurbulence since I can not find it anywhere else in R universe. I would appreciate if you can share the code.<\/p>\n<p>Otherwise, I guess the simplest thing would be to use data from FRED downloading it using getSymbols and using src=”FRED” and the example would be reproducible completely…<\/p>\n<p>Any chance to share code asked for in the comment section of another great article from you <a href=\"http://quantivity.wordpress.com/2012/10/23/volume-clock-gaps-and-goog/\" rel=\"nofollow\">http://quantivity.wordpress.com/2012/10/23/volume-clock-gaps-and-goog/<\/a> ? Thank you.<\/p>\n\t\t\t\t\t\t\t\t\t\t<div class=\"reply\">\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n<\/li>\n\n\t\t\t<li class=\"comment byuser comment-author-quantivity bypostauthor odd alt depth-3 highlander-comment\" id=\"comment-1483\" >\n\t\t\t<div class=\"c-grav\"><img alt='' src='http://1.gravatar.com/avatar/7f67f1b2910b55a3f40a983ffa830357?s=60&d=identicon&r=G' class='avatar avatar-60' height='60' width='60' /><\/div>\n\t\t\t\t<div class=\"c-body\">\n\t\t\t\t\t<div class=\"c-head\">\n\t\t\t\t\t\tquantivity <span class=\"c-permalink\"><a href=\"http://quantivity.wordpress.com/2012/11/09/multi-asset-market-regimes/#comment-1483\">permalink<\/a><\/span><span class=\"asterisk\">*<\/span>\t\t\t\t\t<\/div>\n\t\t\t\t\t<div class=\"c-date\">\n\t\t\t\t\t\tNovember 10, 2012 8:58 am\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<p>@Eduardo: I added dROC to the post. I have a version of the data access functions functions which stub into <code>quantmod<\/code> equivalents (and vice versa, so these functions interoperate with OHLCV functions), which I can make available in a subsequent post depending on reader interest.<\/p>\n<p>Covering turbulence is on the agenda for subsequent post in this series, as this paper depends on it (not to mention Kritzman is an author on a preceding paper covering turbulence).<\/p>\n<p>Finally, I can share TAQ code in a post subsequent to this market regime series.<\/p>\n\t\t\t\t\t\t\t\t\t\t<div class=\"reply\">\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n\n\t\t\t<li class=\"comment even thread-odd thread-alt depth-1 highlander-comment\" id=\"comment-1493\" >\n\t\t\t<div class=\"c-grav\"><img alt='' src='http://1.gravatar.com/avatar/4515b06f9f1c037ccfad149c4c45a88e?s=60&d=identicon&r=G' class='avatar avatar-60' height='60' width='60' /><\/div>\n\t\t\t\t<div class=\"c-body\">\n\t\t\t\t\t<div class=\"c-head\">\n\t\t\t\t\t\tgj <span class=\"c-permalink\"><a href=\"http://quantivity.wordpress.com/2012/11/09/multi-asset-market-regimes/#comment-1493\">permalink<\/a><\/span>\t\t\t\t\t<\/div>\n\t\t\t\t\t<div class=\"c-date\">\n\t\t\t\t\t\tNovember 12, 2012 9:41 pm\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<p>Thanks for the hmmFit package pointer.<\/p>\n\t\t\t\t\t\t\t\t\t\t<div class=\"reply\">\n\t\t\t\t\t\t<a class='comment-reply-link' href='/2012/11/09/multi-asset-market-regimes/?replytocom=1493#respond' onclick='return addComment.moveForm(\"comment-1493\", \"1493\", \"respond\", \"10079\")'>Reply<\/a>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n<\/li>\n\n\t\t\t<li class=\"comment odd alt thread-even depth-1 highlander-comment\" id=\"comment-1505\" >\n\t\t\t<div class=\"c-grav\"><img alt='' src='http://0.gravatar.com/avatar/fde8b319ca36fe6605a65d21f818d2cd?s=60&d=identicon&r=G' class='avatar avatar-60' height='60' width='60' /><\/div>\n\t\t\t\t<div class=\"c-body\">\n\t\t\t\t\t<div class=\"c-head\">\n\t\t\t\t\t\tNiclas <span class=\"c-permalink\"><a href=\"http://quantivity.wordpress.com/2012/11/09/multi-asset-market-regimes/#comment-1505\">permalink<\/a><\/span>\t\t\t\t\t<\/div>\n\t\t\t\t\t<div class=\"c-date\">\n\t\t\t\t\t\tNovember 16, 2012 11:04 pm\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<p>Thank you for a great post. You mention you have a large private data warehouse. Which sources do you use? Any suggestions on how to store such quantity of data (H2, sql, csv, …). Any special R packages you recomend?<\/p>\n<p>Thanks, looking forward to your next post.<\/p>\n\t\t\t\t\t\t\t\t\t\t<div class=\"reply\">\n\t\t\t\t\t\t<a class='comment-reply-link' href='/2012/11/09/multi-asset-market-regimes/?replytocom=1505#respond' onclick='return addComment.moveForm(\"comment-1505\", \"1505\", \"respond\", \"10079\")'>Reply<\/a>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n<\/li>\n\n\t\t\t<li class=\"comment byuser comment-author-7ovevol even thread-odd thread-alt depth-1 highlander-comment\" id=\"comment-1521\" >\n\t\t\t<div class=\"c-grav\"><img alt='' src='http://0.gravatar.com/avatar/990b8e4bb954a47aed9f94395e6c04a7?s=60&d=identicon&r=G' class='avatar avatar-60' height='60' width='60' /><\/div>\n\t\t\t\t<div class=\"c-body\">\n\t\t\t\t\t<div class=\"c-head\">\n\t\t\t\t\t\t<a href='http://convolutedvol' rel='external nofollow' class='url'>Convoluted Volatility<\/a> <span class=\"c-permalink\"><a href=\"http://quantivity.wordpress.com/2012/11/09/multi-asset-market-regimes/#comment-1521\">permalink<\/a><\/span>\t\t\t\t\t<\/div>\n\t\t\t\t\t<div class=\"c-date\">\n\t\t\t\t\t\tNovember 24, 2012 6:11 pm\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<p>Reblogged this on <a href=\"http://convolutedvolatility.com/2012/11/25/multi-asset-market-regimes/\" rel=\"nofollow\">Convoluted Volatility – StatArb,VolArb; Macro.<\/a>.<\/p>\n\t\t\t\t\t\t\t\t\t\t<div class=\"reply\">\n\t\t\t\t\t\t<a class='comment-reply-link' href='/2012/11/09/multi-asset-market-regimes/?replytocom=1521#respond' onclick='return addComment.moveForm(\"comment-1521\", \"1521\", \"respond\", \"10079\")'>Reply<\/a>\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n<\/li>\n\t\t<\/ol>\n\t\t<div class=\"navigation\">\n\t\t<div class=\"alignleft\"><\/div>\n\t\t<div class=\"alignright\"><\/div>\n\t<\/div>\n\t\t<\/div>\n\n\t\t\t\t\t\t\t\t<div id=\"respond\">\n\t\t\t\t<h3 id=\"reply-title\">Leave a Reply <small><a rel=\"nofollow\" 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href=\"http://www.zerohedge.com/\">ZeroHedge<\/a><\/li>\n\n\t<\/ul>\n<\/li>\n<li id=\"twitter-2\" class=\"widget widget_twitter\"><h2 class=\"widgettitle\"><a href='http://twitter.com/quantivity'>Twitter Updates<\/a><\/h2><ul class='tweets'>\n\t\t\t<li>\n\t\t\t\tJoint Training of Deep Boltzmann Machines: <a href=\"http://bit.ly/TXx4a6\"> bit.ly/TXx4a6<\/a> \t\t\t\t<a href=\"http://twitter.com/quantivity/statuses/279270463606247424\" class=\"timesince\">3 hours ago<\/a>\n\t\t\t<\/li>\n\n\t\t\t\n\t\t\t<li>\n\t\t\t\tOnline Portfolio Selection: A Survey: <a href=\"http://bit.ly/TXwXLz\"> bit.ly/TXwXLz<\/a> \t\t\t\t<a href=\"http://twitter.com/quantivity/statuses/279270238233718785\" class=\"timesince\">3 hours ago<\/a>\n\t\t\t<\/li>\n\n\t\t\t\n\t\t\t<li>\n\t\t\t\tTrading Costs of Asset Pricing Anomalies: <a href=\"http://stanford.io/Xf2er8\"> stanford.io/Xf2er8<\/a> \t\t\t\t<a href=\"http://twitter.com/quantivity/statuses/279115236517232640\" class=\"timesince\">14 hours 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