The conditional expectations are estimated using non-parametric kernel regression methods such that , and is a kernel estimator [28]. Equation (3) is then transformed into a well-known linear form: At this point the estimated values of the parameter are obtained using panel data specification estimation methods [34]. Here fixed or random effects panel data econometrics specifications could be applied. The fixed effects specification allows a possible correlation between the variable and the country effects . The random effects specification assumes independence between and the country effects . Since the independence between the individual unobserved effects and the regressors cannot be established ex ante, we use a Hausman specification test [35,36] to test the suitability of the fixed effects assumption. In this test the hypothesis is that the individual unobserved effects and the regressors are uncorrelated . Step 2: Given the estimates of , then estimate the function using the relation By writing , we have the non-parametric form: 917 (3) (4) (5) (6) (7) Int. J. Environ. Res. Public Health 2011, 8 many cases of malaria in remote area are unreported. However, it is well known in the econometric literature [33] that these measurement errors will not affect the consistency of our estimated coefficients. By subtraction of (2) from (1) we have: and is estimated locally minimizing , with being a kernel density function. Ullah and Mundra [37] show that by expressing function such that ), where we can estimate as a smooth coefficient ) as: where (*) implies that local estimation using kernel estimator on conditional expectations is applied. This specification takes into account the fact that climate variability can cause a jump in the number of malaria cases or change the rate of growth in the malaria cases. This particular specification also reduces the dimensionality of the nonparametric function . Li and Racine [28] provide in-depth insight regarding the properties of these estimators. 5. Data Malaria outbreak data were drawn from WHO reports [3,14] for 1990 to 2000 of 25 African countries representing a balanced panel with 319 observations. Associated monthly temperature and precipitation data were drawn from Mitchell et al. [38,39]. Data on demographic and other control variables were assembled as follows: