This thesis discusses the goodness-of-fit testing of an error distribution in a nonparametric autoregressive conditionally heteroscedastic model of order one and in the linear measurement error model.For the nonparametric autoregressive conditionally heteroscedastic model of order one, the test is based on a weighted empirical distribution function of the residuals, where the residuals are obtained from a local linear fit for the autoregressive and heteroscedasticity functions, and the weights are chosen to adjust for the undesirable behavior of these nonparametric estimators in the tails of their domains. An symptotically distribution free test is obtained via Khmaladze martingale transformation. A simulation study is included to assess the finite sample level and power behavior of this test. It exhibits some superiority of this test compared to the classical Kolmogorov-Smirnov and Cram\'er-von Mises tests in terms of the finite sample level and power.For the linear measurement error model, a class of test statistics are based on the integrated square difference between the deconvolution kernel density estimators of the regression model error density and a smoothed version of the null error density, an analog of the so called Bickel and Rosenblatt test statistics. The asymptotic null distributions of the proposed test statistics are derived for both the ordinary smooth and super smooth cases. The asymptotic powers of the proposed tests against a fixed alternative and a class of local nonparametric alternatives for both cases are also described. A finite sample simulation study shows some superiority of the proposed test compared to some other tests.
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