Let’s begin with the ARCH model.<\/p>\n\n\n\n
The ARCH model was initially raised by Engle (1982), and the ARCH model means the A<\/strong>utor<\/strong>egressive C<\/strong>onditional H<\/strong>eteroskedasticity model. <\/p>\n\n\n\n We assume here \\(u_t\\) is the return.<\/p>\n\n\n\n $$ u_t=\\frac{P_t-P_{t-1}}{P_{t-1}} $$<\/p>\n\n\n\n $$u_t\\sim N(0,\\sigma_t^2)$$<\/p>\n\n\n\n The data-generating process (DGP) is like an AR form, as the name of ARCH. The volatility is autoregressively generated by \\(u^2_i\\).<\/p>\n\n\n\n $$\\sigma_t^2=\\delta_0+\\sum_{i=1}^{p} \\delta_i u_{t-i}^2$$<\/p>\n\n\n\n , where \\(p\\) is the number of lags, and \\(\\delta_i\\) are a set of parameters. The DGP of that model shows that the volatility of the return is heteroscedastic, correlated with the squared term of the return per se. <\/p>\n\n\n\n For example, an ARCH(1) model is like,<\/p>\n\n\n\n $$ \\sigma_t^2=\\delta_0+\\delta_1 u^2_{t-1} $$<\/p>\n\n\n\n Note here we need our time series to be stationary for better forecasting. Thus, \\(Var(u_t)=\\sigma^2 \\)<\/p>\n\n\n\n $$ Var(u_t)=\\delta_0+\\delta_1 Var(u_{t-1}) $$<\/p>\n\n\n\n $$ \\sigma^2=\\frac{\\delta_0}{1-\\delta_1} $$<\/p>\n\n\n\n As the variance has to be positive. We need \\(\\delta_0 > 0\\), and \\(\\delta_1<1\\).<\/p>\n\n\n\n For this time series data, OLS assumptions are violated, because our series are autoregressive heteroskedasticity. <\/p>\n\n\n\n Instead, the Maximum Likelihood Estimation (MLE) <\/strong>would be a better estimation method by assuming the probability distribution of variables.<\/p>\n\n\n\n MLE allows iterations to find parameters \\(\\delta\\) that can maximise the maximum likelihood function.<\/p>\n\n\n\n The ‘G’ in the GARCH model means ‘generalised’, and the GARCH model has a set of additional terms, \\(\\sum \\gamma_i \\sigma^2_i \\). Thus, the DGP of the GARCH(p,q) model is as the following,<\/p>\n\n\n\n $$u_t\\sim N(0,\\sigma_t^2)$$<\/p>\n\n\n\n $$ \\sigma_t^2=\\delta_0 + \\sum_{i=1}^{p} \\delta_i u^2_{t-i} +\\sum^q_{j=1} \\gamma_j \\sigma^2_{t-j} $$<\/p>\n\n\n\n That is a further application, in which the GARCH model is applied to mimic the movement of error terms in the ARMA model.<\/p>\n\n\n\n We initially assume an ARMA(p,q) model,<\/p>\n\n\n\n $$ y_t=\\beta_0 +\\sum^p_{i=1} \\beta_i y_{t-i} + \\sum^{q}_{j=1} \\theta_j u_{t-j} +u_t$$<\/p>\n\n\n\n Then, we assume the error term here, \\(u_t \\sim GARCH(m,n)\\).<\/p>\n\n\n\n $$ u_t \\sim N(0,\\sigma_t^2)$$<\/p>\n\n\n\n $$ \\sigma_t^2 = \\delta_0 +\\sum^m_{i=1} \\delta_i u_{t-i}^2 +\\sum_{j=1}^n \\gamma_n \\sigma_{t-n}^2 $$<\/p>\n\n\n\n Engle, R.F., 1982. Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation.\u00a0Econometrica: Journal of the econometric society<\/em>, pp.987-1007.<\/p>\n","protected":false},"excerpt":{"rendered":" Let’s begin with the ARCH model. ARCH Model The ARCH model was initially raised by Engle (1982), and the ARCH model means the Autoregressive Conditional Heteroskedasticity model. We assume here \\(u_t\\) is the return. $$ u_t=\\frac{P_t-P_{t-1}}{P_{t-1}} $$ $$u_t\\sim N(0,\\sigma_t^2)$$ The data-generating process (DGP) is like an AR form, as the name of ARCH. The volatility … Continue reading ARCH and GARCH<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[14,6],"tags":[],"_links":{"self":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/4478"}],"collection":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=4478"}],"version-history":[{"count":63,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/4478\/revisions"}],"predecessor-version":[{"id":4541,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=\/wp\/v2\/posts\/4478\/revisions\/4541"}],"wp:attachment":[{"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=4478"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=4478"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/fanyuzhao.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=4478"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}GARCH Model<\/h4>\n\n\n\n
ARMA-GARCH Model<\/h4>\n\n\n\n
Reference<\/h4>\n\n\n\n