Nonlinear and State-Dependent Local Projections
Tenreyro and Thwaites (2016)
This chapter reproduces the nonlinear local-projection designs in Tenreyro and Thwaites (2016) with fNLLP(). The original MATLAB archive estimates a single US quarterly time series. It is neither a panel model nor LP-IV: the identified monetary shock is placed directly in each projection. Three designs appear in the archive:
- a smooth-transition state-dependent LP (
stlpm.m), - positive versus negative shocks (
stlpm2.m), and - a linear-plus-cubic shock response (
stlpm3.m).
The baseline uses horizons 0 through 20, one lag of the outcome, one lag of the federal funds rate, a common linear trend, and 90% intervals. Its expansion weight is a logistic transform with slope 3 around the 20th percentile of trailing seven-quarter GDP growth. The workbook supplied with the archive contains that transition weight, so the replication uses it directly.
1 The three specifications
Let \(s_t\) be the identified monetary shock, \(x_t\) the controls, and \(M_t\in[0,1]\) the expansion weight.
The sign-asymmetric projection is
\[y_{t+h}=\alpha_h+\beta_h^+\max(s_t,0)+\beta_h^-\min(s_t,0)+\gamma_h'x_t+u_{t+h}.\]
fNLLP() reports positive as \(\beta_h^+\) and negative as \(-\beta_h^-\). Both therefore describe a shock of unit magnitude: a \(+1\) tightening and a \(-1\) loosening.
The cubic projection is
\[y_{t+h}=\alpha_h+\beta_{1h}s_t+\beta_{3h}s_t^3+\gamma_h'x_t+u_{t+h}.\]
The response to a shock of size \(d\) is \(\beta_{1h}d+\beta_{3h}d^3\). predict() evaluates this expression and its HAC variance, including the covariance between the linear and cubic coefficients.
The smooth state-dependent projection is
\[y_{t+h}=\delta_h t +M_t\left(\alpha_h^E+\beta_h^E s_t+\gamma_h^{E\prime}x_t\right) +(1-M_t)\left(\alpha_h^R+\beta_h^R s_t+\gamma_h^{R\prime}x_t\right) +u_{t+h}.\]
This matches stlpm.m: the trend coefficient is common, while the intercept, shock, and controls receive expansion and recession coefficients. fNLLP() estimates both coefficients in one regression and returns their HAC covariance, so the standard error for \(\beta_h^E-\beta_h^R\) is valid.
The rest will come here….