# Dynamic Linear Quantile Regression Interface to `rq.fit` and `rq.wfit` for fitting dynamic linear quantile regression models. The interface is based very closely on Achim Zeileis's dynlm package. In effect, this is mainly ``syntactic sugar'' for formula processing, but one should never underestimate the value of good, natural sweeteners. ```r dynrq(formula, tau = 0.5, data, subset, weights, na.action, method = "br", contrasts = NULL, start = NULL, end = NULL, ...) ``` ## Arguments - `formula`: a `"formula"` describing the linear model to be fit. For details see below and `rq`. - `tau`: the quantile(s) to be estimated, may be vector valued, but all all values must be in (0,1). - `data`: an optional `"data.frame"` or time series object (e.g., `"ts"` or `"zoo"`), containing the variables in the model. If not found in `data`, the variables are taken from `environment(formula)`, typically the environment from which `rq` is called. - `subset`: an optional vector specifying a subset of observations to be used in the fitting process. - `weights`: an optional vector of weights to be used in the fitting process. If specified, weighted least squares is used with weights `weights` (that is, minimizing `sum(w*e^2)`); otherwise ordinary least squares is used. - `na.action`: a function which indicates what should happen when the data contain `NA`s. The default is set by the `na.action` setting of `options`, and is `na.fail` if that is unset. The factory-fresh default is `na.omit`. Another possible value is `NULL`, no action. Note, that for time series regression special methods like `na.contiguous`, `na.locf` and `na.approx` are available. - `method`: the method to be used; for fitting, by default `method = "br"` is used; `method = "fn"` employs the interior point (Frisch-Newton) algorithm. The latter is advantageous for problems with sample sizes larger than about 5,000. - `contrasts`: an optional list. See the `contrasts.arg` of `model.matrix.default`. - `start`: start of the time period which should be used for fitting the model. - `end`: end of the time period which should be used for fitting the model. - ``...``: additional arguments to be passed to the low level regression fitting functions. ## Details The interface and internals of `dynrq` are very similar to `rq`, but currently `dynrq` offers two advantages over the direct use of `rq` for time series applications of quantile regression: extended formula processing, and preservation of time series attributes. Both features have been shamelessly lifted from Achim Zeileis's package dynlm. For specifying the `formula` of the model to be fitted, there are several functions available which allow for convenient specification of dynamics (via `d()` and `L()`) or linear/cyclical patterns (via `trend()`, `season()`, and `harmon()`). These new formula functions require that their arguments are time series objects (i.e., `"ts"` or `"zoo"`). Dynamic models: An example would be `d(y) ~ L(y, 2)`, where `d(x, k)` is `diff(x, lag = k)` and `L(x, k)` is `lag(x, lag = -k)`, note the difference in sign. The default for `k` is in both cases `1`. For `L()`, it can also be vector-valued, e.g., `y ~ L(y, 1:4)`. Trends: `y ~ trend(y)` specifies a linear time trend where `(1:n)/freq` is used by default as the covariate, `n` is the number of observations and `freq` is the frequency of the series (if any, otherwise `freq = 1`). Alternatively, `trend(y, scale = FALSE)` would employ `1:n` and `time(y)` would employ the original time index. Seasonal/cyclical patterns: Seasonal patterns can be specified via `season(x, ref = NULL)` and harmonic patterns via `harmon(x, order = 1)`. `season(x, ref = NULL)` creates a factor with levels for each cycle of the season. Using the `ref` argument, the reference level can be changed from the default first level to any other. `harmon(x, order = 1)` creates a matrix of regressors corresponding to `cos(2 * o * pi * time(x))` and `sin(2 * o * pi * time(x))` where `o` is chosen from `1:order`. See below for examples. Another aim of `dynrq` is to preserve time series properties of the data. Explicit support is currently available for `"ts"` and `"zoo"` series. Internally, the data is kept as a `"zoo"` series and coerced back to `"ts"` if the original dependent variable was of that class (and no internal `NA`s were created by the `na.action`). ## See Also `zoo`, `merge.zoo` ## Examples ```r ########################### # Models ## ########################### if(require(zoo)){ ## multiplicative median SARIMA(1,0,0)(1,0,0)_12 model fitted to UK seatbelt data uk <- log10(UKDriverDeaths) dfm <- dynrq(uk ~ L(uk, 1) + L(uk, 12)) dfm dfm3 <- dynrq(uk ~ L(uk, 1) + L(uk, 12),tau = 1:3/4) summary(dfm3) ## explicitly set start and end dfm1 <- dynrq(uk ~ L(uk, 1) + L(uk, 12), start = c(1975, 1), end = c(1982, 12)) ## remove lag 12 dfm0 <- update(dfm1, . ~ . - L(uk, 12)) tuk1 <- anova(dfm0, dfm1) ## add seasonal term dfm1 <- dynrq(uk ~ 1, start = c(1975, 1), end = c(1982, 12)) dfm2 <- dynrq(uk ~ season(uk), start = c(1975, 1), end = c(1982, 12)) tuk2 <- anova(dfm1, dfm2) ## regression on multiple lags in a single L() call dfm3 <- dynrq(uk ~ L(uk, c(1, 11, 12)), start = c(1975, 1), end = c(1982, 12)) anova(dfm1, dfm3) } ############################### ## Time Series Decomposition ## ############################### ## airline data ## Not run: ap <- log(AirPassengers) fm <- dynrq(ap ~ trend(ap) + season(ap), tau = 1:4/5) sfm <- summary(fm) plot(sfm) ## End(Not run) ## Alternative time trend specifications: ## time(ap) 1949 + (0, 1, ..., 143)/12 ## trend(ap) (1, 2, ..., 144)/12 ## trend(ap, scale = FALSE) (1, 2, ..., 144) ############################### ## An Edgeworth (1886) Problem## ############################### # DGP ## Not run: fye <- function(n, m = 20){ a <- rep(0,n) s <- sample(0:9, m, replace = TRUE) a[1] <- sum(s) for(i in 2:n){ s[sample(1:20,1)] <- sample(0:9,1) a[i] <- sum(s) } zoo::zoo(a) } x <- fye(1000) f <- dynrq(x ~ L(x,1)) plot(x,cex = .5, col = "red") lines(fitted(f), col = "blue") ## End(Not run) ```

dynrq function