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I'm trying to estimate the maximum likelihood of a realized GARCH model. Below are the equations and the parameters I want to estimate

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I'm using the below function to maximise the likelihood, but it has being very inconsistent. Moreover, it is highly sensitive to the initial parameters, it always spits out value very very close to the initial parameters. I've tried using real data and also simulated data, so I could check against the true parameters. Am I doing anything wrong in the code? I know that GARCH models usually have a flat likelihood but still I thought I would get some different values from my initial ones

 garch11_realised <- function (returns, params, RM) { #Starting our parameters omega <- params[1] beta <- params[2] gamma <- params[3] xi <- params[4] phi <- params[5] tau1 <- params[6] tau2 <- params[7] T <- length(returns) log_sigma2 <- numeric(T) sigma2 <- numeric(T) log_x <- log(RM) ut <- numeric(T) z <- numeric(T) tau_z <- numeric(T) log_sigma2[1] <- log((omega + gamma * xi) / (1 - (beta + gamma*phi))) sigma2[1] <- exp(log_sigma2[1]) z[1] <- returns[1] / sqrt(exp(log_sigma2[1])) for (t in 2:T){ #we are using log here because that's how the model is specified in most of the papers log_sigma2[t] <- omega + beta * log_sigma2[t-1] + gamma * log_x[t-1] sigma2[t] <- exp(log_sigma2[t]) z[t] <- returns[t] / sqrt(exp(log_sigma2[t])) tau_z[t] <- tau1 *z[t] + tau2 * (z[t] ^ 2 - 1) #our leverage function ut[t]<- log_x[t] - xi - phi * log_sigma2[t] - tau_z[t] # rearranging measurement equation to calculate the residuals } return(list(sigma2 = sigma2, ut = ut, z = z)) } neg_loglik_realised <- function(returns, params, RM){ omega <- params[1] beta <- params[2] gamma <- params[3] xi <- params[4] phi <- params[5] tau1 <- params[6] tau2 <- params[7] sigma_ut <- params[8] #variance of our ut in the measurement equation T <- length(returns) model_results <- garch11_realised(returns,params, RM) sigma2 <- model_results$sigma2 ut <- model_results$ut #required conditions for realized garch - https://core.ac.uk/download/pdf/41239655.pdf - page 6 of this paper # 0 < β + φγ < 1 and ω + γξ > 0 if (all(is.finite(params)) && omega > 0 && beta > 0 && gamma > 0 && 0 < (beta + phi * gamma) && (beta + phi * gamma) < 1 && (omega + gamma * xi) > 0) { #quasi log-likelihood log_lik_1 <- - T/2* log(2*pi) - 1/2 * sum(log(sigma2)) - 1/2 * sum(returns^2 / sigma2) #this is the first part l(r) log_lik_2 <- - T/2 * log(2 * pi) - 1/2 * sum(log(sigma_ut)) -1/2 * sum((ut ^ 2 / sigma_ut ^ 2)) #this is the second part l(x|r) neg_lik <- - (log_lik_1 + log_lik_2) } else { neg_lik <- Inf } return(neg_lik) } estimating_mle_realised <- function(start_params,returns, RM) { lower_bounds = c(1e-8, 1e-8, 1e-8, -1, -1, -1, -1, -1) upper_bounds = c(1, 0.99, 0.99, 1, 99, 1, 1, 1) mle <- nlminb(start = start_params, objective = function(start_params) objective_fn(start_params,returns, RM), lower = lower_bounds, upper = upper_bounds) return(mle) }

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  • $\begingroup$ Try adding a counter to your if ... else statement at the bottom, to see how often your optimizer searches outside the parameter space and returns Inf. $\endgroup$ Commented May 12, 2024 at 20:05
  • $\begingroup$ @Durden, thanks. it really depends on the lower and upper bounds I set in my estimating_mle function. Depending on the bounds I set, I get 6 results outside the parameter space or as high as 83... I am trying to control for this within my garch function by adding the below: ' numerator = omega + gamma * xi denominator = 1 - (beta + gamma * phi) if (numerator > 0 && denominator > 0) { log_sigma2[1] <- log(numerator / denominator) } else { log_sigma2[1] <- log(var(returns)) # Fallback to variance of returns } ' $\endgroup$ Commented May 13, 2024 at 8:41
  • $\begingroup$ Hi: I realize that you want to do it yourself but there's an R package called rugarch. If you don't want to use it directly, maybe you could view the source and see if there are subtleties that you are missing as far as when you move out of the admissible parameter space. You just need to download the package from cran and either tar it or unzip the binary if you're on windows. $\endgroup$ Commented May 13, 2024 at 10:48
  • $\begingroup$ Hi, thanks. Yes, I have already fitted a realized model with the package, however, I want to do it myself because I will change the GARCH specification to incorporate a HAR structure, which it isn't possible through the package. Also, I've been trying to look through their source code for the past 2 days but can't really match my results with theirs... $\endgroup$ Commented May 13, 2024 at 11:08
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    $\begingroup$ I don't have time to look through the Rugarch source code but what I've done in the past when I had a similar problem ( not Rugarch ) is call the function in Rugarch something else and copy it to your R environment Then, step through it in the R-debugger to see what's happening at each step of the calculation. That might be the only way. What you're trying to do is not easy so I wouldn't get discouraged. $\endgroup$ Commented May 13, 2024 at 15:30

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