Lecture 12 Heteroscedasticity 5

GARCH: Forecasting and Persistence Example 1 (continuation): We forecast volatility for March 2021: 0.00012 ∗ {1 + (0.9) + (0.9)2+ … + (0.9)5} + 𝐸 : 𝜎 : + 0.00367 ∗ (0.9)6 = 0.002512659 Remark: We observe that as the forecast horizon increases (j → ∞), the forecast reverts to the unconditional variance: 𝜔/(1 – α1 – β1) = 0.00012/(1 − 0.9) = 0.0012  𝜎 = sqrt(0.0012) = 0.0346 (3.46% ≈ close to sample SD = 3.36%) 97 GARCH: Forecasting and Persistence Example 2: On August 2020, we forecast the December’s variance for the S&P500 changes. Recall we estimated 𝜎 : 𝜎 = 0.756 + 0.125 𝜀 + 0.826 𝜎 . getting 𝜎 : = 43.037841 We based the 𝜎 𝐸 𝜎 Then, since 𝜶𝟏 𝐸 : 𝜎 : : forecast on: 𝜔∗ ∑ 𝛼 𝛽 𝜎 𝛼 𝛽 𝜷𝟏 = 0.952 0.756 ∗ {1+ (0.952) + (0.952)2 + (0.952)3} + + 43.037841 ∗ (0.952)4 = 38.02797 Lower variance forecasted for the end of the year, but still far from the unconditional variance of 15.4. 98 49 ARCH: Which Model to Use • Questions 1) Lots of ARCH models. Which one to use? 2) Choice of p and q. How many lags to use? • Hansen and Lunde (2004) compared lots of ARCH models: - It turns out that the GARCH(1, 1) is a great starting model. - Add a leverage effect for financial series and it’s even better. - A t-distribution is also a good addition. RV Models: Intuition • The idea of realized volatility is to estimate the latent (unobserved) variance using the realized data, without any modeling. Recall the definition of sample variance: 𝒔𝟐 𝑻 1 𝑇 1 𝒙𝒊 𝒙 𝟐 • Suppose we want to calculate the daily variance for stock returns. We know how to compute it: we use daily information, for T days, and apply the above definition. • Alternatively, we use hourly data for the whole day (with 𝑘 hours). Since hourly returns are very small, ignoring 𝒙 seems OK. We use 𝑟 , as the 𝑖 th hourly variance on day 𝑡. Then, we add 𝑟 , over the day: 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 𝑟, 10 0 50 RV Models: Intuition • In more general terms, we use higher frequency data to estimate a lower frequency variance: 𝑅𝑉 𝑟 ,𝒊 where 𝑟 , is the realized returns in (higher frequency) interval 𝑖 of the (lower frequency) period 𝑡. We estimate the 𝑡-frequency variance, using 𝑘 𝑖-intervals. If we have daily returns and we want to estimate the monthly variance, then, 𝑘 is equal to the number of days in a month. • It can be shown that as the interval 𝑖 becomes smaller (𝑖 → 0), 𝑅𝑉 → Return Variation [𝑡 1, 𝑡]. That is, with an increasing number of observations we get an accurate measure of the latent variance. 10 1 RV Models: High Frequency • Note that RV is a model-free measure of variation –i.e., no need for ARCH-family specifications. The measure is called realized variance (RV). The square root of the realized variance is the realized volatility (RVol, RealVol): 𝑅𝑉𝑜𝑙 𝑠𝑞𝑟𝑡 𝑅𝑉 • Given the previous theoretical result, RV is commonly used with intra-daily data, called high frequency (HF) data. • It lead to a revolution in the field of volatility, creating new models and new ways of thinking about volatility and how to model it. • We usually associate realized volatility with an observable proxy of the unobserved volatility. 10 2 51 RV Models: High Frequency – Tick Data • The theory behind realized variation measures dictates that the sampling frequency, or 𝑘 in the 𝑅𝑉 formula above, goes to ∞. Then, use highest frequency available, say millisecond to millisecond returns. • Intra-daily data applications are the most common. But, when using intra-daily data, RV calculations are affected by microstructure effects: bidask bounce, infrequent trading, calendar effects, etc. 𝑟 , does not look uncorrelated. Example: The bid-ask bounce induces serial correlation in intra-day returns, which biases 𝑅𝑉 . • As the sampling frequency increases, the “noise” (microstructure effects) becomes more dominant and swallows the “signal” (true volatility). 10 3 RV Models: High Frequency – Tick Data • In practice, sampling a typical stock price every few seconds can overestimate the true volatility by a factor of two or more. • The usual solutions: (1) Filter data using an ARMA model to get rid of the autocorrelations and/or dummy variables to get rid of calendar effects. Then, used the filtered data to compute 𝑅𝑉 . (2) Sample at frequencies where the impact of microstructure effects is minimized and/or eliminated. We follow solution (2). 10 4 52 RV Models: High Frequency – Practice • In intra-daily RV estimation, it is common to use 10’ intervals. They have good properties. However, there are estimations with 1’ intervals. • Some studies suggest using an optimal frequency, where optimal frequency is the one that minimizes the MSE. • Hansen and Lunde (2006) find that for very liquid assets, such as the S&P 500 index, a 5’ sampling frequency provides a reasonable choice. Thus, to calculate daily RV, we need to add 78 five-minute intervals. 10 5 RV Models: High Frequency – TAQ Example: Based on TAQ (Trade and Quote) NYSE data, we use 5’ realized returns to calculate 30’ variances –i.e., we use six 5’ intervals. , is: Then, the 30’ variance, or 𝑅𝑉 𝑅𝑉 ∑ 𝑟, , 𝑡 1, 2, . . . . , 𝑇 = 13 𝑟 , is the 5’ return during the 𝑗th interval on the half hour 𝑡. Then, we calculate 30’ variances for the whole day –i.e., we calculate 13 variances, since the trading day goes from 9:30 AM to 4:00 PM. The Realized Volatility, 𝑅𝑉𝑜𝑙, is: 𝑅𝑉𝑜𝑙 𝑅𝑉 10 6 53 RV Models: High Frequency – TAQ Example: Below, we show the first transaction of the SPY TAQ (Trade and Quote) data (tick-by-tick trade data) on January 2, 2014. SYMBOL DATE TIME PRICE SIZE SPY 20140102 9:30:00 183.98 500 SPY 20140102 9:30:00 183.98 500 SPY 20140102 9:30:00 183.98 200 SPY 20140102 9:30:00 183.98 500 SPY 20140102 9:30:00 183.98 1000 SPY 20140102 9:30:00 183.98 1000 SPY 20140102 9:30:00 183.98 800 SPY 20140102 9:30:00 183.98 100 SPY 20140102 9:30:00 183.98 100 SPY 20140102 9:30:00 183.97 200 SPY 20140102 9:30:00 183.98 100 SPY 20140102 9:30:00 183.97 200 SPY 20140102 9:30:00 183.98 1000 SPY 20140102 9:30:00 183.97 100 SPY 20140102 9:30:00 183.98 1000 SPY 20140102 9:30:00 183.98 2600 SPY 20140102 9:30:00 183.98 1000 SPY 20140102 9:30:00 183.97 400 10 7 RV Models: High Frequency – TAQ Example: Below, we show the first transaction of the AAPL TAQ (Trade and Quote) data (tick-by-tick quote data) on January 2, 2014: 4 AM SYMBOL DATE AAPL 20140102 TIME 4:00:00 BID OFR AAPL 20140102 4:00:00 553.5 558 2 2 12 P AAPL 20140102 4:00:01 455.39 561.02 1 2 12 T AAPL 20140102 4:00:45 552.1 558 1 2 12 P AAPL 20140102 4:00:51 552.1 558.4 1 2 12 P AAPL 20140102 4:00:51 552.1 558.8 1 2 12 P AAPL 20140102 4:00:51 552.1 559 1 1 12 P AAPL 20140102 4:01:14 553 559 1 1 12 P AAPL 20140102 4:01:30 553.01 561.02 1 2 12 T AAPL 20140102 4:01:43 553.01 559 1 1 12 T AAPL 20140102 4:01:44 553.05 559 1 1 12 P AAPL 20140102 4:01:49 455.39 559 1 1 12 T AAPL 20140102 4:01:49 553.61 559 1 1 12 T AAPL 20140102 4:02:02 553.05 559 1 2 12 P AAPL 20140102 4:02:04 455.39 559 1 1 12 T AAPL 20140102 4:02:04 548.28 559 1 1 12 T AAPL 20140102 4:02:33 553.05 558.83 1 2 12 P AAPL 20140102 4:02:33 555.17 558.83 2 2 12 P AAPL 20140102 4:03:50 555.2 558.83 5 2 12 P 455.39 BIDSIZ 0 OFRSIZ 1 MODE 0 EX 12 T 10 8 54 RV Models: High Frequency – TAQ Example (continuation): We read SPY trade data for 2014:Jan. > HF_da <- read.csv("c:/Financial Econometrics/SPY_2014.csv", head=TRUE, sep=",") > summary(HF_da) SYMBOL DATE SPY:6800865 Min. :20140102 1st Qu.:20140110 Median :20140121 Mean :20140119 3rd Qu.:20140128 Max. :20140131 TIME PRICE SIZE G127 9:30:00 : 21436 Min. :176.6 Min. : 1 Min. :0 16:00:00: 11352 1st Qu.:178.9 1st Qu.: 100 1st Qu.:0 9:30:01 : 5922 Median :182.6 Median : 100 Median :0 15:59:59: 4090 Mean :181.4 Mean : 337 Mean :0 15:59:55: 3198 3rd Qu.:183.5 3rd Qu.: 300 3rd Qu.:0 15:50:00: 2916 Max. :189.2 Max. :4715350 Max. :0 (Other) :6751951 CORR COND EX Min. :0.0e+00 @ :3351783 T :1649158 1st Qu.:0.0e+00 F :2888182 P :1335135 Median :0.0e+00 : 524409 Z :1182126 Mean :1.9e-04 O : 18057 D :1062382 3rd Qu.:0.0e+00 4 : 9098 K : 437900 Max. :1.2e+01 6 : 8142 J : 356539 (Other): 1194 (Other): 777625 10 9 RV Models: High Frequency – TAQ Example (continuation): Using the SPY trade data, we calculate using 5’-returns a daily realized volatility for the first 4 days in 2014 (2014:01:02 - 2014:01:07). Originally, we have T = 1,048,570. HF_da <- read.csv("http://www.bauer.uh.edu//rsusmel//4397//SPY_2014.csv", head=TRUE, sep=",") summary(HF_da) pt <- as.POSIXct(paste(HF_da$DATE, HF_da$TIME), format="%Y%m%d %H:%M:%S") library(xts) hf_1 <- xts(x=HF_da, order.by = pt) spy_p <- as.numeric(hf_1$PRICE) # Define a specific time series data set # pt pastes together DATE and Time. # Read price data as numeric T <- length(spy_5_p) spy_ret <- log(spy_p[-1]/spy_p[-T]) plot(spy_ret, type="l", ylab="Return", main="Tick by Tick Return (2014:01:02 - 2014:01:07)") mean(spy_ret) sd(spy_ret) 11 0 55 RV Models: High Frequency – TAQ Example (continuation): We plot the tick-by-tick data. Very noisy data, with lots of “jumps”: Mean tick by tick return: -3.7365e-09 Tick-by-tick SD: 6.3163e-05 111 RV Models: High Frequency – TAQ Example (continuation): For the whole month of January 2014: > mean(spy_ret) [1] -4.796933e-09 > sd(spy_ret) [1] 7.804991e-05 11 2 56 RV Models: High Frequency – TAQ Example (continuation): We plot the autocorrelogram for the TAQ SPY data: > acf_spy_raw Autocorrelations of series ‘spy_ret’, by lag 0 1 2 3 4 5 6 7 8 9 10 1.000 -0.469 -0.013 -0.010 0.014 -0.008 0.000 -0.002 -0.001 0.000 0.000 Note: We have only a significant autocorrelation, the 1st-order autocorrelation: -0.459. 11 3 RV Models: High Frequency – TAQ Example (continuation): We aggregate the tick-by-tick data in 5’ intervals using the function aggregateTrades in the R package highfrequency. It needs as an input an xts object (hf_1, for us). library(highfrequency) spy_5 <- aggregateTrades( hf_1, on = "minutes", k = 5, marketOpen = "09:30:00", marketClose = "16:00:00", tz = "GMT" ) # you can use also seconds, days, weeks, etc. # number of units in for “on” spy_5_p <- as.numeric(spy_5$PRICE) T <- length(spy_5_p) spy_5_ret <- log(spy_5_p[-1]/spy_5_p[-T]) plot(spy_5_ret, type="l", ylab="Return", main="5-minute Return (2014:01:02 - 2014:01:07)")11 4 57 RV Models: High Frequency – TAQ Example (continuation): We plot the 5-minute return data. Smoother, easier to read. RVolt=2014:01:02 = 0.0053344 RVolt=2014:01:03 = 0.0043888 RVolt=2014:01:04 = 0.0059836 RVolt=2014:01:05 = 0.0052772 11 5 RV Models: High Frequency – TAQ Example (continuation): We plot the autocorrelogram for the 5’ TAQ SPY data: > acf_spy_5 <- acf(spy_5_ret, main = "5-minute SPY Data: January 2014") > acf_spy_5 Autocorrelations of series ‘spy_ret’, by lag 0 1 2 3 4 5 6 7 8 9 10 1.000 -0.105 -0.024 -0.104 0.018 0.147 0.016 -0.024 -0.088 0.048 0.037 Note: We have a negative 1st-order autocorrelation: -0.105, thought not 11 significant. However, the autocorrelation of order 5 is significant. 6 58 RV Models: High Frequency – TAQ Example (continuation): We plot the 10-minute return data. Smoothing increases. RVolt=2014:01:02 = 0.005478294 RVolt=2014:01:03 = 0.004256046 RVolt=2014:01:04 = 0.006190508 RVolt=2014:01:05 = 0.005145601 11 7 RV Models: High Frequency – TAQ Example (continuation): We plot the autocorrelogram for the 10’ TAQ SPY data: Note: Now, none of the autocorrelations is significant. The 10-minute returns look independent. 11 8 59 RV Models: R Script Example: R script to compute realized volatility MSCI_da <- read.csv("http://www.bauer.uh.edu/rsusmel/4397/MSCI_daily.csv", head=TRUE, sep=",") x_us <- MSCI_da$USAT <- length(x_us) us_r <- log(x_us[-1]/x_us[-T]) x <- us_r T <- length(x) rvs=NULL i <- 1 k <- 21 while (i < T-k) { s2 <- sum(x[i:(i+k)]^2) i <- k + i rvs <- rbind(rvs,s2) } rvol <- sqrt(rvs) mean(rvol) sd(rvol) # US log returns from MSCI USA Index # create vector to fill with RV # k: observations per period # realized variance # realized volatility # mean # variance RV Models: Monthly RV From Daily Data Example: Using daily data we calculate 1-mo Realized Volatility (𝑘 = 21 days) for log returns for the MSCI (1970: Jan – 2020: Oct). > mean(rvol) [1] 0.04326531 > sd(rvol) [1] 0.02592653 # average monthly Rvol in the sample  very close to monthly S&P Volatility: 4.49% # standard deviation of monthly Rvol in the sample  dividing by sqrt(T) we get the SE = 0.001 (very small) 60