# Estimation of truncated regression models using the Left Truncated (LT) estimator Estimates linear regression models with truncated response variables (fixed truncation point), using the LT estimator (Karlsson 2006). ```r lt( formula, data, point = 0, direction = "left", clower = "ml", const = 1, cupper = 2, beta = "ml", covar = FALSE, na.action, ... ) ``` ## Arguments - `formula`: a symbolic description of the model to be estimated - `data`: an optional data frame - `point`: the value of truncation (the default is 0) - `direction`: the direction of truncation, either `"left"` (the default) or `"right"` - `clower`: the lower threshold value to be used when trimming the conditional density of the errors from below. The default is `"ml"` meaning that the residual standard deviation from fitting a maximum likelihood model for truncated regression, using `truncreg`, is used. Method `"ols"` uses the estimated residual standard deviation from a linear model fitted by `lm`. It is also possible to manually supply the threshold value by setting `clower` to be equal to a number or numeric vector of length one. - `const`: a number that can be used to alter the size of the lower threshold. `const=0.5` would give a lower threshold value that is half the original size. The default value is 1. - `cupper`: number indicating what upper threshold to use when trimming the conditional density of the errors from above. The number is used to multiply the lower threshold value, i.e. if `cupper=2` (the default value) the upper threshold value is two times larger than the lower threshold value. - `beta`: the method of determining the starting values of the regression coefficients (See Details for more information): * The default method is `"ml"`, meaning that the estimated regression coefficients from fitting a maximum likelihood model for truncated regression, assuming Gaussian errors, are used. The maximum likelihood model is fitted using `truncreg`. * Method `"ols"` means that the estimated regression coefficients from fitting a linear model with `lm` are used. * The third option is to manually provide starting values as either a vector, column matrix or row matrix. - `covar`: logical. Indicates whether or not the covariance matrix should be estimated. If `TRUE` the covariance matrix is estimated using bootstrap. The default number of replicates is 2000 but this can be adjusted (see argument `...`). However, since the bootstrap procedure is time-consuming the default is `covar=FALSE`. - `na.action`: a function which indicates what should happen when the data contain `NA`s. - `...`: additional arguments. For `lt` the number of bootstrap replicates can be adjusted by setting `R=`the desired number of replicates. Also the `control` argument of `optim` can be set by `control=list()` (see Details for more information). ## Returns `lt` returns an object of class `"lt"`. The function `summary` prints a summary of the results, including two types of confidence intervals (normal approximation and percentile method). The generic accessor functions `coef`, `fitted`, `residuals` and `vcov` extract various useful features of the value returned by `lt` An object of class `"lt"`, a list with elements: - **coefficients**: the named vector of coefficients - **startcoef**: the starting values of the regression coefficients used by `optim` - **cvalues**: information about the thresholds used. The method and constant used and the resulting lower and upper threshold values. - **value**: the value of the objective function corresponding to `coefficients` - **counts**: number of iterations used by `optim`. See the documentation for `optim` for further details - **convergence**: from `optim`. An integer code. 0 indicates successful completion. Possible error codes are 1 indicating that the iteration limit maxit had been reached. 10 indicating degeneracy of the Nelder--Mead simplex. - **message**: from `optim`. A character string giving any additional information returned by the optimizer, or `NULL`. - **residuals**: the residuals of the model - **fitted.values**: the fitted values - **df.residual**: the residual degrees of freedom - **call**: the matched call - **covariance**: if `covar = TRUE`, the estimated covariance matrix - **R**: if `covar = TRUE`, the number of bootstrap replicates - **bootrepl**: if `covar = TRUE`, the bootstrap replicates ## Details Minimizes the objective function described in Karlsson (2006) wrt the vector of regression coefficients, in order to find the LT estimates. The minimization is performed by `optim` using the "Nelder--Mead" method, and a maximum number of iterations of 2000. The maximum number of iterations can be adjusted by setting `control = list(maxit = ...)` (for more information see the documentation for `optim`). It is recommended to use one of the methods for generating the starting values of the regression coefficients (see argument `beta`) rather than supplying these manually, unless one is confident that one has a good idea of what these should be. This because the starting values can have a great impact on the result of the minimization. Note that setting `cupper = 1` means that the LT estimates will coincide with the estimates from the Quadratic Mode Estimator (see function `qme`). For more detailed information see Karlsson and Lindmark (2014). ## Examples ```r #################################### # Simulated data # #################################### #Simulated data (asymmetrically distributed errors): n <- 1000 set.seed(319993) x1 <- runif(n, 0, 10) x2 <- runif(n, 0, 10) x3 <- runif(n, -5, 5) eps <- rexp(n, 0.2) - 5 y <- 2 - 2*x1 + x2 + 2*x3 + eps d <- data.frame(y = y, x1 = x1, x2 = x2, x3 = x3) ##Use a truncated subsample dtrunc <- subset(d, y > 0) ##Use lt to consistently estimate the slope parameters lt.sim <- lt(y ~ x1 + x2 + x3, dtrunc, point = 0, direction = "left", clower = "ml", const = 1, cupper = 2, beta = "ml", covar = FALSE) summary(lt.sim) #################################### # Example using data "PM10trunc" # #################################### data(PM10trunc) ltpm10 <- lt(PM10 ~ cars + temp + wind.speed + temp.diff + wind.dir + hour + day, data = PM10trunc, point = 2, control = list(maxit=2500)) summary(ltpm10) ``` ## References Karlsson, M. (2006) Estimators of regression parameters for truncated and censored data, **Metrika**, 63 , pp 329--341 Karlsson, M., Lindmark, A. (2014) truncSP: An R Package for Estimation of Semi-Parametric Truncated Linear Regression Models, **Journal of Statistical Software**, 57(14) , pp 1--19, [https://www.jstatsoft.org/article/view/v057i14](https://www.jstatsoft.org/article/view/v057i14) ## See Also `lt.fit`, the function that does the actual fitting `qme`, for estimation of models with truncated response variables using the QME estimator `stls`, for estimation of models with truncated response variables using the STLS estimator `truncreg` for estimating models with truncated response variables by maximum likelihood, assuming Gaussian errors ## Author(s) Anita Lindmark and Maria Karlsson

lt function