Appendix E: Differential Equations Cheat Sheet

E.1 First-Order ODEs¶

General form: $\dot{x} = f(x, t)$.

Linear first-order (constant coefficients): $\dot{x} = ax + b$.

• Steady state: $x^* = -b/a$ (if $a\neq0$).

• General solution: $x(t) = (x_0 - x^*)e^{at} + x^*$.

• Stable iff $a < 0$.

Stability: Linearize $\dot{x} = f(x)$ at $x^*$: $\dot{u} \approx f'(x^*)u$. Stable iff $f'(x^*) < 0$.

E.2 Second-Order ODEs¶

Linear, constant coefficients: $\ddot{x} + p\dot{x} + qx = r$.

Characteristic equation: $\lambda^2 + p\lambda + q = 0$, roots $\lambda_{1,2} = (-p \pm \sqrt{p^2-4q})/2$.

where $\alpha = -p/2$, $\beta = \sqrt{4q-p^2}/2$, $x^* = r/q$ (particular solution).

Stability: All trajectories → $x^*$ iff $\text{Re}(\lambda_i) < 0$ for both roots.

E.3 Systems of ODEs¶

Linear system: $\dot{\mathbf{x}} = A\mathbf{x} + \mathbf{b}$.

• Steady state: $\mathbf{x}^* = -A^{-1}\mathbf{b}$ (if $A$ invertible).

• General solution: $\mathbf{x}(t) = e^{At}(\mathbf{x}_0 - \mathbf{x}^*) + \mathbf{x}^*$.

• Stable iff all eigenvalues of $A$ have negative real parts.

• Saddle point iff $\det(A) < 0$ (eigenvalues of mixed sign).

Phase portrait classification (2D system, eigenvalues $\lambda_1, \lambda_2$):

E.4 Pontryagin Conditions Summary¶

For $\max_{u(t)}\int_0^\infty e^{-\rho t}F(x,u)dt$ s.t. $\dot{x} = f(x,u)$:

Key costate interpretation: $\mu(t) = \partial V/\partial x$ (shadow price of state = marginal value of relaxing the constraint).

Appendix F: Difference Equations Cheat Sheet¶

F.1 First-Order Difference Equations¶

Linear: $x_{t+1} = ax_t + b$.

• Steady state: $x^* = b/(1-a)$ ($a\neq1$).

• General solution: $x_t = a^t(x_0-x^*) + x^*$.

• Stable iff $|a| < 1$.

Stability and behavior:

Forward-looking equation: $x_t = a\mathbb{E}_t[x_{t+1}] + b z_t$.

• If $|a|<1$: unique bounded solution $x_t = b\sum_{j=0}^\infty a^j\mathbb{E}_t[z_{t+j}]$.

• If $|a|>1$: MSV solution $x_t = b/(1-a\rho_z)z_t$ plus possible sunspots.

F.2 Second-Order Difference Equations¶

Linear: $x_{t+2} + px_{t+1} + qx_t = c$.

Characteristic equation: $\lambda^2 + p\lambda + q = 0$, roots $\lambda_{1,2} = (-p\pm\sqrt{p^2-4q})/2$.

Stability conditions: $|a| < 1+q$ AND $q < 1$ (Schur–Cohn conditions — necessary and sufficient).

Oscillation period (complex case): $T = 2\pi/\theta$ periods.

F.3 Systems of Difference Equations¶

Linear system: $\mathbf{x}_{t+1} = A\mathbf{x}_t + \mathbf{b}$.

• Steady state: $\mathbf{x}^* = (I-A)^{-1}\mathbf{b}$.

• General solution: $\mathbf{x}_t = A^t(\mathbf{x}_0-\mathbf{x}^*)+\mathbf{x}^*$.

• Stable iff all eigenvalues of $A$ satisfy $|\lambda_i| < 1$.

• IRF at horizon $h$: $A^{h-1}\mathbf{b}$ (response to unit impulse).

Companion form. Convert $p$-th order scalar system to first-order system with companion matrix:

Eigenvalues of $A$ are the roots of the characteristic polynomial.

F.4 Blanchard–Kahn Counting Rule¶

For the linearized DSGE $\Gamma_0\mathbf{y}_t = \Gamma_1\mathbf{y}_{t-1}+\Psi\mathbf{z}_t+\Pi\boldsymbol\eta_t$:

F.5 MSV Solution Procedure Summary¶

For $\mathbf{y}_t = A\mathbb{E}_t[\mathbf{y}_{t+1}] + C\mathbf{z}_t$ with $\mathbf{z}_t = \Phi\mathbf{z}_{t-1} + \boldsymbol\varepsilon_t$:

1. Conjecture: $\mathbf{y}_t = \Omega\mathbf{z}_t$.

2. Substitute: $\Omega\mathbf{z}_t = A\Omega\Phi\mathbf{z}_t + C\mathbf{z}_t$.

3. Sylvester equation: $\Omega - A\Omega\Phi = C$.

4. Vectorize: $\text{vec}(\Omega) = (I-\Phi'\otimes A)^{-1}\text{vec}(C)$.

5. Verify: All eigenvalues of $A$ outside unit circle iff unique bounded solution.

Appendix G: Time Series Concepts Summary¶

G.1 Stationarity¶

Strict stationarity. $\{y_t\}$ is strictly stationary if the joint distribution of $(y_{t_1},\ldots,y_{t_k})$ equals that of $(y_{t_1+h},\ldots,y_{t_k+h})$ for all $k$, $h$, and time indices.

Covariance (weak) stationarity. $\mathbb{E}[y_t] = \mu$ (constant), $\text{Cov}(y_t, y_{t-j}) = \gamma_j$ (depends only on lag $j$).

Key distinction: Strict $\Rightarrow$ weak stationarity (for finite variance processes). Weak $\not\Rightarrow$ strict.

Integrated processes. $y_t \sim I(d)$: $d$-th difference is stationary. Most macro levels: $I(1)$; growth rates: $I(0)$.

G.2 ARMA Representations¶

AR($p$): $y_t = \sum_{j=1}^p\phi_jy_{t-j} + \varepsilon_t$. Stationary iff all roots of $1-\phi_1z-\cdots-\phi_pz^p$ lie outside the unit circle.

MA($q$): $y_t = \sum_{j=0}^q\theta_j\varepsilon_{t-j}$. Always stationary.

ARMA($p,q$): Combines both. Autocovariance: $\gamma_j = \text{Cov}(y_t, y_{t-j})$ depends on $j$ only.

Wold decomposition. Any zero-mean $I(0)$ process: $y_t = \sum_{j=0}^\infty\psi_j\varepsilon_{t-j}$ (MA($\infty$)) with $\sum\psi_j^2<\infty$. Long-run multiplier: $\psi(1) = \sum\psi_j$.

G.3 VAR Models¶

VAR($p$): $\mathbf{y}_t = \mathbf{c}+\sum_{j=1}^pA_j\mathbf{y}_{t-j}+\mathbf{e}_t$.

OLS estimation: $\hat{B} = (X'X)^{-1}X'Y$ — APL: B ← (⌹ X) +.× Y.

Lag selection criteria:

BIC consistent; AIC better for forecasting.

Granger causality. $x_t$ Granger-causes $y_t$ iff $A_{yx,j}\neq0$ for some lag $j$ — test via F-test on the restricted vs. unrestricted VAR equations.

G.4 Structural VAR Identification¶

Identification schemes (additional restrictions beyond $\Sigma = B_0^{-1}(B_0^{-1})'$):

IRF: Response of variable $i$ to shock $j$ at horizon $h$: $(A^h_{comp}\cdot B_0^{-1})_{ij}$.

FEVD: Fraction of $h$-step MSE of variable $i$ due to shock $j$: $\sum_{k=0}^{h-1}(\Psi_k B_0^{-1})_{ij}^2/\text{MSE}_i(h)$.

G.5 Cointegration¶

Engle–Granger two-step:

1. OLS: $\hat\beta = (X'X)^{-1}X'Y$ (super-consistent, converges at rate $T$).

2. ADF test on residuals $\hat{u}_t = y_t - \hat\beta x_t$.

Johansen trace statistic: $\Lambda_{trace}(r) = -T\sum_{i=r+1}^n\ln(1-\hat\lambda_i)$.

Error correction (VECM): $\Delta\mathbf{y}_t = \alpha\beta'\mathbf{y}_{t-1}+\sum_{j=1}^{p-1}\Gamma_j\Delta\mathbf{y}_{t-j}+\boldsymbol\varepsilon_t$, where $\beta$ = cointegrating vectors, $\alpha$ = adjustment speeds.

G.6 Key Test Statistics¶