{smcl}
{* *! version 1.1.1}{...}
{title:Title}

{phang}
{bf:npss} {hline 2} Executes nonparametric estimation of heteroskedastic state space models.


{marker syntax}{...}
{title:Syntax}

{p 4 17 2}
{cmd:npss}
{it:y1}
{it:y2}
{ifin}
[{cmd:,} {bf:skedastic}({it:varname}) {bf:tp1}({it:real}) {bf:tp2}({it:real})]


{marker description}{...}
{title:Description}

{phang}
{cmd:npss} executes nonparametric estimation of conditionally heteroskedastic state space models based on 
{browse "https://www.sciencedirect.com/science/article/abs/pii/S0304407617302427":Botosaru and Sasaki (2018)}. 
Consider a state space model {it:y}({it:it}) = {it:u}({it:it}) + {it:v}({it:it}), where {it:y}({it:it}) is observed (e.g., earnings), {it:u}({it:it}) is unobserved (e.g., permanent component of earnings), and {it:v}({it:it}) is unobserved (e.g., transitory component of earnings), with the process {it:u}({it:it}) = {it:u}({it:it-1}) + {it:w}({it:it}). Taking {it:y}({it:i1}) and {it:y}({it:i2}) as input, the command nonparametrically estimates and draws the density functions of {it:u}({it:i1}) and {it:v}({it:i1}). Taking {it:y}({it:i1}), {it:y}({it:i2}) and {it:y}({it:i3}) as input, the command also nonparametrically estimates and draws the conditional skedastic function of {it:u}({it:i2}) given {it:u}({it:i1}), e.g., as a measure of heterogeneous risks in permanent component of earnings.


{marker options}{...}
{title:Options}

{phang}
{bf:skedastic({it:varname})} tells the command to estimate the skedastic function of {it:u}({it:i2}) given {it:u}({it:i1}). The input in this option is {bf:y3}, the observed variable in the third time period after the first two, {bf:y1} and {bf:y2}. Not calling this option tells the command to estimate only the density functions of {it:u}({it:i1}) and {it:v}({it:i1}).

{phang}
{bf:tp1({it:real})} sets the scale-normalized tuning parameter for estimation of the density functions. The default value is {bf: tp1(4)}.

{phang}
{bf:tp2({it:real})} sets the scale-normalized tuning parameter for estimation of the skedastic function. The default value is {bf: tp2(2)}.


{marker examples}{...}
{title:Examples}

{phang}
({bf:y2006}, {bf:y2008}, & {bf:y2010}: earnings in 2006, 2008, & 2010, respectively.){p_end}

{phang}Estimation of the density functions of {it:u}({it:2006}) and {it:v}({it:2006}), using {bf:y2006} and {bf:y2008} as input:

{phang}{cmd:. use "example_2006_2008_2010.dta"}{p_end}
{phang}{cmd:. npss y2006 y2008}{p_end}

{phang}Estimation the conditional skedastic function of {it:u}({it:2008}) given {it:u}({it:2006}), in addition to the density functions of {it:u}({it:2006}) and {it:v}({it:2006}), using {bf:y2006}, {bf:y2008} and {bf:y2010} as input:

{phang}{cmd:. use "example_2006_2008_2010.dta"}{p_end}
{phang}{cmd:. npss y2006 y2008, skedastic(y2010)}{p_end}


{marker stored}{...}
{title:Stored results}

{phang}
{bf:npss} stores the following in {bf:r()}: 
{p_end}

{phang}
Scalars
{p_end}
{phang2}
{bf:r(N)} {space 10}observations
{p_end}

{phang}
Macros
{p_end}
{phang2}
{bf:r(cmd)} {space 8}{bf:npss}
{p_end}

{phang}
Matrices
{p_end}
{phang2}
{bf:r(U)} {space 10}density f(U) of U
{p_end}
{phang2}
{bf:r(V)} {space 10}density f(V) of V
{p_end}
{phang2}
{bf:r(S)} {space 10}conditional skedastic function sigma(U)
{p_end}


{title:Reference}

{p 4 8}Botosaru, I. and Y. Sasaki. 2018. Nonparametric Heteroskedasticity in Persistent Panel Processes: An Application to Earnings Dynamics. {it:Journal of Econometrics}, 203 (2), pp. 283-296. 
{browse "https://www.sciencedirect.com/science/article/abs/pii/S0304407617302427":Link to Paper}.
{p_end}


{title:Authors}

{p 4 8}Irene Botosaru, University of Bristol, Bristol, UK.{p_end}

{p 4 8}Yuya Sasaki, Vanderbilt University, Nashville, TN.{p_end}