diff --git a/day2/live_code_examples/live_3_timevarying.R b/day2/live_code_examples/live_3_timevarying.R
index b1967cc5..d2c926fe 100644
--- a/day2/live_code_examples/live_3_timevarying.R
+++ b/day2/live_code_examples/live_3_timevarying.R
@@ -1,143 +1,195 @@
-library(mgcv)
-library(dplyr)
-library(marginaleffects)
-library(gratia)
-library(ggplot2); theme_set(theme_bw())
-
-#### Time-varying coefficients in mgcv ####
-
-# First a function to simulate from a squared exponential Gaussian Process
-# N: number of timepoints to simulate
-# c: constant (average coefficient value)
-# alpha: amplitude of variation
-# rho: length scale (how 'wiggly' should the time-varying effect be?)
-sim_gp = function(N, c, alpha, rho){
-  Sigma <- alpha ^ 2 *
-           exp(-0.5 * ((outer(1:N, 1:N, "-") / rho) ^ 2)) +
-           diag(1e-9, N)
- c + mgcv::rmvn(1,
-                mu = rep(0, N),
-                V = Sigma)
-}
-
-# Simulate a time-varying coefficient that has a mean value of 0.5
-# (i.e. the effect tends to be positive, but can change smoothly over time)
-set.seed(3)
-N <- 150
-beta_t <- sim_gp(alpha = 0.75,
-               rho = 15,
-               c = 0.5,
-               N = N)
-
-# Plot the coefficient
-plot(beta_t, type = 'l', lwd = 3,
-     bty = 'l', xlab = 'Time',
-     ylab = 'Coefficient',
-     col = 'darkred')
-
-# Now simulate the covariate itself
-x <- rnorm(N, 5, sd = 1)
-plot(x, type = 'l', lwd = 3,
-     bty = 'l', xlab = 'Time',
-     ylab = 'Covariate (x)',
-     col = 'darkred')
-
-# Simulate the linear predictor, which is a simple linear model:
-# mu = beta_0 + x_t * beta_t
-beta_0 <- 3
-mu <- beta_0 + x * beta_t
-
-# And simulate some Gaussian noise
-y <- rnorm(N, mean = mu, sd = 0.5)
-plot(y, type = 'l', lwd = 3,
-     bty = 'l', xlab = 'Time',
-     ylab = 'Outcome (y)',
-     col = 'darkred')
-
-# Fit a model that assumes the coefficient is static
-dat <- data.frame(y, x, time = 1:N)
-mod0 <- gam(y ~ x + time, data = dat, method = 'REML')
-summary(mod0)
-plot_predictions(mod0, condition = 'x', points = 0.5)
-plot_predictions(mod0, by = 'time', points = 0.5)
-avg_slopes(mod0, variables = 'x')
-appraise(mod0, n_simulate = 100, method = 'simulate')
-
-# Now a model that allows the coefficient to vary over time;
-# this makes use of the 'by' argument in the s() function. The idea is
-# that we fit a smooth function of 'time' that gets multiplied with the covariate
-# x, effectively allowing us to estimate arbitrarily-nonlinear effects that change
-# over time
-mod1 <- gam(y ~ s(time, by = x, k = 50),
-            data = dat, method = 'REML')
-summary(mod1)
-draw(mod1)
-plot_predictions(mod1, by = c('time'), points = 0.5)
-
-summary_round = function(x){
-  round(fivenum(x, na.rm = TRUE), 4)
-}
-plot_predictions(mod1, by = c('time', 'x'),
-                 newdata = datagrid(x = summary_round,
-                                    time = 1:N),
-                 points = 0.5)
-
-# Easier to visualise the time-varying effect if we fix the x value
-mean_round = function(x){
-  round(mean(x, na.rm = TRUE), 4)
-}
-plot_predictions(mod1, by = c('time', 'x'),
-                 newdata = datagrid(x = mean_round,
-                                    time = 1:N))
-
-avg_slopes(mod1, variables = 'x')
-appraise(mod1, n_simulate = 100, method = 'simulate')
-
-# How would this smooth extrapolate?
-plot_predictions(mod1, by = c('time', 'x'),
-                 newdata = datagrid(x = mean_round,
-                                    time = 1:(N + 20)))
-
-# Not really very realistic; what we want is a proper time series
-# model for the coefficient. A quick way to do this in mgcv is to use
-# a Random Walk basis
-library(MRFtools) # devtools::install_github("eric-pedersen/MRFtools")
-rw_penalty <- mrf_penalty(object = 1:(N + 20),
-                          type = 'linear')
-dat$time_factor <- factor(1:N, levels = as.character(1:(N + 20)))
-dat$time_factor
-
-mod2 <- gam(y ~ s(time_factor, by = x,
-                  bs = "mrf",
-                  k = 25,
-                  xt = list(penalty = rw_penalty)),
-            data = dat,
-            drop.unused.levels = FALSE,
-            method = "REML")
-summary(mod2)
-draw(mod2)
-plot_predictions(mod2, by = c('time_factor', 'x'),
-                 newdata = datagrid(x = mean_round,
-                                    time_factor = 1:(N + 20)))
-
-# Because 'time' is a factor here, it is a bit harder to visualise. It is probably
-# easier to resort to our own plots
-newdat <- datagrid(model = mod2,
-                   x = mean_round,
-                   time_factor = 1:(N + 20)) %>%
-  # Add a continuous version of 'time' for plotting
-  dplyr::mutate(time = 1:(N + 20))
-preds <- predict(mod2,
-                 newdata = newdat,
-                 type = 'response', se = TRUE)
-newdat$pred <- preds$fit
-newdat$upper <- preds$fit + 1.96*preds$se.fit
-newdat$lower <- preds$fit - 1.96*preds$se.fit
-ggplot(newdat, aes(x = time, y = pred)) +
-  geom_line(linewidth = 1, alpha = 0.6) +
-  geom_ribbon(aes(ymin = lower, ymax = upper),
-              alpha = 0.3, col = NA) +
-  theme_classic() +
-  theme(legend.position = 'none')
-
-# These extrapolations are much more realistic!
+library(mgcv)
+library(dplyr)
+library(marginaleffects)
+library(gratia)
+library(ggplot2); theme_set(theme_bw())
+
+# Define a function to simulate from a squared exponential Gaussian Process
+# N: number of timepoints to simulate
+# c: constant (average coefficient value)
+# alpha: amplitude of variation
+# rho: length scale (how 'wiggly' should the time-varying effect be?)
+sim_gp = function(N, c, alpha, rho){
+  Sigma <- alpha ^ 2 *
+    exp(-0.5 * ((outer(1:N, 1:N, "-") / rho) ^ 2)) +
+    diag(1e-9, N)
+  c + mgcv::rmvn(1,
+                 mu = rep(0, N),
+                 V = Sigma)
+}
+
+# Simulate a time-varying coefficient that has a mean value of 0.5
+# (i.e. the effect tends to be positive, but can change smoothly over time)
+set.seed(3)
+N <- 150
+beta_t <- sim_gp(alpha = 0.75,
+                 rho = 15,
+                 c = 0.5,
+                 N = N)
+
+# Plot the coefficient
+plot(beta_t, type = 'l', lwd = 3,
+     bty = 'l', xlab = 'Time',
+     ylab = 'Coefficient',
+     col = 'darkred')
+
+# Now simulate the covariate itself
+x <- rnorm(N, 5, sd = 1)
+plot(x, type = 'l', lwd = 3,
+     bty = 'l', xlab = 'Time',
+     ylab = 'Covariate (x)',
+     col = 'darkred')
+
+# Simulate the linear predictor, which is a simple linear model:
+# mu = beta_0 + x_t * beta_t
+beta_0 <- 3
+mu <- beta_0 + x * beta_t
+
+# And simulate some Gaussian noise
+y <- rnorm(N, mean = mu, sd = 0.5)
+plot(y, type = 'l', lwd = 3,
+     bty = 'l', xlab = 'Time',
+     ylab = 'Outcome (y)',
+     col = 'darkred')
+
+# Fit a linear model that assumes the coefficient is static
+dat <- data.frame(y, x, time = 1:N)
+mod0 <- gam(y ~ x + time, data = dat, method = 'REML')
+summary(mod0)
+
+# Use plot_predictions() for effect interrogation
+plot_predictions(mod0, condition = 'x', points = 0.5)
+plot_predictions(mod0, by = 'time', points = 0.5)
+
+# We can also calculate the average slope for the effect of x
+avg_slopes(mod0, variables = 'x')
+
+# Model diagnostics with gratia
+appraise(mod0, n_simulate = 100, method = 'simulate')
+
+# This model clearly is inadequate. 
+# Now try a model that allows the coefficient to vary over time;
+# this makes use of the 'by' argument in the s() function. The idea is
+# that we fit a smooth function of 'time' that gets multiplied with the covariate
+# x, effectively allowing us to estimate arbitrarily-nonlinear effects that change
+# over time (look at the documentation on the 'by' argument in the s() help page
+# for more details)
+?mgcv::s
+mod1 <- gam(y ~ s(time, by = x, k = 25),
+            data = dat, method = 'REML')
+summary(mod1)
+
+# Now we can use draw() from gratia to inspect the estimated smooth
+draw(mod1)
+
+# And again use plot_predictions() to interrogate effects
+summary_round = function(x){
+  round(fivenum(x, na.rm = TRUE), 4)
+}
+plot_predictions(mod1, by = c('time', 'x'),
+                 newdata = datagrid(x = summary_round,
+                                    time = 1:N),
+                 points = 0.5)
+
+# It may be easier to visualise the time-varying effect if we fix the x value
+mean_round = function(x){
+  round(mean(x, na.rm = TRUE), 4)
+}
+plot_predictions(mod1, by = c('time', 'x'),
+                 newdata = datagrid(x = mean_round,
+                                    time = 1:N))
+
+# Average slope is again easy to compute
+avg_slopes(mod1, variables = 'x')
+
+# And model diagnostics look better now
+appraise(mod1, n_simulate = 100, method = 'simulate')
+
+# But how would this smooth extrapolate if we wanted to forecast ahead?
+plot_predictions(mod1, by = c('time', 'x'),
+                 newdata = datagrid(x = mean_round,
+                                    time = 1:(N + 20))) +
+  geom_vline(xintercept = N, linetype = 'dashed')
+
+# Not really very realistic, the exrapolation behaviour is completely different
+# from the historical behaviour; what we want is a proper time series
+# model for the coefficient. A quick way to do this in mgcv is to use
+# a Random Walk basis
+library(MRFtools) # devtools::install_github("eric-pedersen/MRFtools")
+?mgcv::mrf
+
+# Setting up a RW penalty requires a factor version of time, with factor 
+# levels that extend as far ahead as we would like to forecast
+rw_penalty <- mrf_penalty(object = 1:(N + 20),
+                          type = 'linear')
+dat$time_factor <- factor(1:N, levels = as.character(1:(N + 20)))
+dat$time_factor
+
+# Fit the model using the mrf basis and including the RW penalty
+mod2 <- gam(y ~ s(time_factor, by = x,
+                  bs = "mrf",
+                  k = 25,
+                  xt = list(penalty = rw_penalty)),
+            data = dat,
+            
+            # Keep all factor levels in the data
+            drop.unused.levels = FALSE,
+            method = "REML")
+summary(mod2)
+
+# draw() doesn't work yet for MRFs
+draw(mod2)
+
+# plot_predictions() will work, but the result isn't what we're after
+plot_predictions(mod2, by = c('time_factor', 'x'),
+                 newdata = datagrid(x = mean_round,
+                                    time_factor = 1:(N + 20))) +
+  geom_vline(xintercept = N, linetype = 'dashed')
+
+# Because 'time' is a factor here, it is a bit harder to visualise. It is probably
+# easier to resort to our own plots. Create a datagrid() for predicting over
+newdat <- datagrid(model = mod2,
+                   x = mean_round,
+                   time_factor = 1:(N + 20)) %>%
+  # Add a continuous version of 'time' for plotting
+  dplyr::mutate(time = 1:(N + 20))
+
+# Predict from the model using this grid
+preds <- predict(mod2,
+                 newdata = newdat,
+                 type = 'response', se = TRUE)
+newdat$pred <- preds$fit
+newdat$upper <- preds$fit + 1.96*preds$se.fit
+newdat$lower <- preds$fit - 1.96*preds$se.fit
+
+# Visualise the predictions
+ggplot(newdat, aes(x = time, y = pred)) +
+  geom_line(linewidth = 1, alpha = 0.6, col = 'darkred') +
+  geom_ribbon(aes(ymin = lower, ymax = upper),
+              alpha = 0.3, col = NA, fill = 'darkred') +
+  theme_classic() +
+  theme(legend.position = 'none')
+
+# These extrapolations are much more realistic
+
+#### Bonus tasks ####
+# 1. Plot residuals of mod2 against the predictor x, and against time, to 
+#    look for any unmodelled autocorrelation
+
+# 2. Compare the mgcv models (mod0, mod1 and mod2)
+#    Generalized Likelihood Ratio test (hint, see anova.gam() for help)
+?anova.gam
+
+# 3. If you have mvgam installed, you can go one better using the dynamic() wrapper
+#    to model the temporal effect with a Gaussian Process
+library(mvgam)
+mod3 <- mvgam(y ~ dynamic(x, k = 25, scale = FALSE),
+              data = dat,
+              family = gaussian())
+
+#    Use mcmc_plot() to inspect posterior estimates for the GP parameters,
+#    and compare them to the simulated truths
+?mcmc_plot.mvgam
+
+#    Use plot_predictions() as you did for mod1 to inspect how the time-varying
+#    effect extrapolates