fix equations in documentation

nbs/docs/models/ARCH.ipynb

@@ -83,8 +83,9 @@
     "\n",
     "Specifically, we give the definition of the ARCH model as follows. \n",
     "\n",
-    "**Definition 1.**  An $\\text{ARCH(p)}$ model with order $p≥1$ is ofthe form\n",
+    "**Definition 1.**  An $\\text{ARCH(p)}$ model with order $p≥1$ is of the form\n",
     "\n",
+    "$$\n",
     "\\begin{equation}\n",
     "    \\left\\{\n",
     "\t    \\begin{array}{ll}\n",
@@ -93,6 +94,7 @@
     "\t    \\end{array}\n",
     "\t\\right.\n",
     "\\end{equation}\n",
+    "$$\n",
     "\n",
     "where $\\omega ≥ 0, \\alpha_i ≥ 0$, and $\\alpha_p > 0$ are constants, $\\varepsilon_t \\sim iid(0, 1)$, and $\\varepsilon_t$ is independent of $\\{X_k;k ≤ t − 1 \\}$. A stochastic process $X_t$ is called an $ARCH(p)$ process if it satisfies Eq. (1).\n",
     "\n",
@@ -101,11 +103,17 @@
     "Let $\\mathscr{F}_s$ denote the information set generated by $\\{X_k;k ≤ s \\}$, namely, the sigma field $\\sigma(X_k;k ≤ s)$. It is easy to see that $\\mathscr{F}_s$ is independent of $\\varepsilon_t$ for any $s <t$. According to Definition 1 and the properties of the conditional mathematical\n",
     "expectation, we have that\n",
     "\n",
-    "$$E(X_t|\\mathscr{F}_{t−1}) = E(\\sigma_t \\varepsilon_t|\\mathscr{F}_{t−1}) = \\sigma_t E( \\varepsilon_t|\\mathscr{F}_{t−1}) = \\sigma_t E(\\varepsilon_t) = 0 \\tag 2$$\n",
+    "$$\n",
+    "\\begin{equation}\n",
+    "    E(X_t|\\mathscr{F}_{t−1}) = E(\\sigma_t \\varepsilon_t|\\mathscr{F}_{t−1}) = \\sigma_t E( \\varepsilon_t|\\mathscr{F}_{t−1}) = \\sigma_t E(\\varepsilon_t) = 0 \\tag 2\n",
+    "\\end{equation}\n",
+    "$$\n",
     "\n",
     "and\n",
     "\n",
-    "$$ \\text{Var}(X_{t}^2| \\mathscr{F}_{t−1}) = E(X_{t}^2|\\mathscr{F}_{t−1}) =  E(\\sigma_{t}^2 \\varepsilon_{t}^2|\\mathscr{F}_{t−1}) = \\sigma_{t}^2 E(\\varepsilon_{t}^2|\\mathscr{F}_{t−1}) = \\sigma_{t}^2 E(\\varepsilon_{t}^2) = \\sigma_{t}^2. $$\n",
+    "$$\n",
+    "\\text{Var}(X_{t}^2| \\mathscr{F}_{t−1}) = E(X_{t}^2|\\mathscr{F}_{t−1}) =  E(\\sigma_{t}^2 \\varepsilon_{t}^2|\\mathscr{F}_{t−1}) = \\sigma_{t}^2 E(\\varepsilon_{t}^2|\\mathscr{F}_{t−1}) = \\sigma_{t}^2 E(\\varepsilon_{t}^2) = \\sigma_{t}^2.\n",
+    "$$\n",
     "\n",
     "This implies that $\\sigma_{t}^2$ is the conditional variance of $X_t$ and it evolves according to the previous values of $\\{X_{k}^2; t −p ≤ k ≤ t −1\\}$ like an $\\text{AR}(p)$ model. And so Model (1) is named an $\\text{ARCH}(p)$ model.\n"
    ]
@@ -116,6 +124,7 @@
    "source": [
     "As an example of $\\text{ARCH}(p)$ models, let us consider the $\\text{ARCH(1)}$ model\n",
     "\n",
+    "$$\n",
     "\\begin{equation}\n",
     "    \\left\\{ \n",
     "\t    \\begin{array}{ll} \\tag 3\n",
@@ -124,16 +133,22 @@
     "\t    \\end{array}\n",
     "\t\\right.\n",
     "\\end{equation}\n",
+    "$$\n",
     "\n",
     "Explicitly, the unconditional mean\n",
     "$$E(X_t) = E(\\sigma_t \\varepsilon_t) = E(\\sigma_t) E(\\varepsilon_t) = 0. $$\n",
     "\n",
     "Additionally, the ARCH(1) model can be expressed as\n",
     "\n",
-    "$$X_{t}^2 =\\sigma_{t}^2 +X_{t}^2 − \\sigma_{t}^2  =\\omega +\\alpha_1 X_{t-1}^2 +\\sigma_{t}^2 \\varepsilon_{t}^2 −\\sigma_{t}^2  =\\omega +\\alpha_1 X_{t}^2 +\\eta_t ,$$\n",
+    "$$X_{t}^2 =\\sigma_{t}^2 +X_{t}^2 − \\sigma_{t}^2  =\\omega +\\alpha_1 X_{t-1}^2 +\\sigma_{t}^2 \\varepsilon_{t}^2 −\\sigma_{t}^2  =\\omega +\\alpha_1 X_{t}^2 +\\eta_t$$\n",
     "\n",
     "that is,\n",
-    "$$X_{t}^2 =\\omega +\\alpha_1 X_{t}^2 +\\eta_t \\tag 4 $$\n",
+    "\n",
+    "$$\n",
+    "\\begin{equation}\n",
+    "    X_{t}^2 =\\omega +\\alpha_1 X_{t}^2 +\\eta_t \\tag 4\n",
+    "\\end{equation}\n",
+    "$$\n",
     "\n",
     "where $\\eta_t = \\sigma_{t}^2(\\varepsilon_{t}^2 − 1)$. It can been shown that $\\eta_t$ is a new white noise, which is left as an exercise for reader. Hence, if $0 < \\alpha_1 < 1$, Eq. (4) is a stationary $\\text{AR(1)}$ model for the series Xt2. Thus, the unconditional variance\n",
     "\n",
@@ -196,7 +211,7 @@
    "source": [
     "### Autoregressive Conditional Heteroskedasticity (ARCH) Applications\n",
     "\n",
-    "* **Finance** – The ARCH model is widely used in finance to model volatility in financial time series, such as stock prices, exchange rates, interest rates, etc.\n",
+    "* **Finance** - The ARCH model is widely used in finance to model volatility in financial time series, such as stock prices, exchange rates, interest rates, etc.\n",
     "\n",
     "* **Economics** - The ARCH model can be used to model volatility in economic data, such as GDP, inflation, unemployment, among others.\n",
     "\n",

nbs/docs/models/ARIMA.ipynb

@@ -101,13 +101,13 @@
     "   - d is the number of differencing required to make the time series stationary\n",
     "\n",
     "\n",
-    "- **AR(p) Autoregression** – a regression model that utilizes the dependent relationship between a current observation and observations over a previous period. An auto regressive (AR(p)) component refers to the use of past values in the regression equation for the time series.\n",
+    "- **AR(p) Autoregression** - a regression model that utilizes the dependent relationship between a current observation and observations over a previous period. An auto regressive (AR(p)) component refers to the use of past values in the regression equation for the time series.\n",
     "\n",
     "\n",
-    "- **I(d) Integration** – uses differencing of observations (subtracting an observation from observation at the previous time step) in order to make the time series stationary. Differencing involves the subtraction of the current values of a series with its previous values d number of times.\n",
+    "- **I(d) Integration** - uses differencing of observations (subtracting an observation from observation at the previous time step) in order to make the time series stationary. Differencing involves the subtraction of the current values of a series with its previous values d number of times.\n",
     "\n",
     "\n",
-    "- **MA(q) Moving Average** – a model that uses the dependency between an observation and a residual error from a moving average model applied to lagged observations. A moving average component depicts the error of the model as a combination of previous error terms. The order q represents the number of terms to be included in the model.\n",
+    "- **MA(q) Moving Average** - a model that uses the dependency between an observation and a residual error from a moving average model applied to lagged observations. A moving average component depicts the error of the model as a combination of previous error terms. The order q represents the number of terms to be included in the model.\n",
     "\n",
     "\n"
    ]
@@ -173,18 +173,23 @@
     "\n",
     "Thus, an autoregressive model of order p can be written as\n",
     "\n",
-    "$$y_{t} = c + \\phi_{1}y_{t-1} + \\phi_{2}y_{t-2} + \\dots + \\phi_{p}y_{t-p} + \\varepsilon_{t}, \\tag{1}$$\n",
-    "\n",
-    "\n",
+    "$$\n",
+    "\\begin{equation}\n",
+    "y_{t} = c + \\phi_{1}y_{t-1} + \\phi_{2}y_{t-2} + \\dots + \\phi_{p}y_{t-p} + \\varepsilon_{t} \\tag{1}\n",
+    "\\end{equation}\n",
+    "$$\n",
     "\n",
-    "where $\\epsilon_t$ is white noise. This is like a multiple regression but with lagged values of  $y_t$ as predictors. We refer to this as an AR( p) model, an autoregressive model of order p\n",
-    " .\n",
+    "where $\\epsilon_t$ is white noise. This is like a multiple regression but with lagged values of  $y_t$ as predictors. We refer to this as an AR( p) model, an autoregressive model of order p.\n",
     "\n",
     "\n",
     "### MA model\n",
     "Rather than using past values of the forecast variable in a regression, a moving average model uses past forecast errors in a regression-like model,\n",
     "\n",
-    "$$y_{t} = c + \\varepsilon_t + \\theta_{1}\\varepsilon_{t-1} + \\theta_{2}\\varepsilon_{t-2} + \\dots + \\theta_{q}\\varepsilon_{t-q}, \\tag{2}$$\n",
+    "$$\n",
+    "\\begin{equation}\n",
+    "y_{t} = c + \\varepsilon_t + \\theta_{1}\\varepsilon_{t-1} + \\theta_{2}\\varepsilon_{t-2} + \\dots + \\theta_{q}\\varepsilon_{t-q} \\tag{2}\n",
+    "\\end{equation}\n",
+    "$$\n",
     "\n",
     "where  $\\epsilon_t$ is white noise. We refer to this as an MA(q) model, a moving average model of order q. Of course, we do not observe the values of  \n",
     "$\\epsilon_t$ , so it is not really a regression in the usual sense.\n",
@@ -233,11 +238,11 @@
     "autoregressive model|\n",
     "|ARIMA (0, 1, 1)|0   1   1 |$\\hat Y_t = Y_{t-1} - \\Phi_1 e^{t-1}$|Simple exponential\n",
     "smoothing|\n",
-    "|ARIMA (0, 0, 1)|0   0   1 |$\\hat Y_t = \\mu_0+ \\epsilon_t – \\omega_1 \\epsilon_{t-1}$|MA(1): First-order\n",
+    "|ARIMA (0, 0, 1)|0   0   1 |$\\hat Y_t = \\mu_0+ \\epsilon_t - \\omega_1 \\epsilon_{t-1}$|MA(1): First-order\n",
     "regression model|\n",
-    "ARIMA (0, 0, 2) |0   0   2 |$\\hat Y_t = \\mu_0+ \\epsilon_t – \\omega_1 \\epsilon_{t-1} – \\omega_2 \\epsilon_{t-2}$|MA(2): Second-order\n",
+    "ARIMA (0, 0, 2) |0   0   2 |$\\hat Y_t = \\mu_0+ \\epsilon_t - \\omega_1 \\epsilon_{t-1} - \\omega_2 \\epsilon_{t-2}$|MA(2): Second-order\n",
     "regression model|\n",
-    "|ARIMA (1, 0, 1)|1   0   1 |$\\hat Y_t = \\Phi_0 + \\Phi_1 Y_{t-1}+ \\epsilon_t – \\omega_1 \\epsilon_{t-1}$|ARMA model|\n",
+    "|ARIMA (1, 0, 1)|1   0   1 |$\\hat Y_t = \\Phi_0 + \\Phi_1 Y_{t-1}+ \\epsilon_t - \\omega_1 \\epsilon_{t-1}$|ARMA model|\n",
     "|ARIMA (1, 1, 1)|1   1   1 |$\\Delta Y_t = \\Phi_1 Y_{t-1} + \\epsilon_t - \\omega_1 \\epsilon_{t-1}$ |ARIMA model|\n",
     "|ARIMA (1, 1, 2)|1   1   2 |$\\hat Y_t = Y_{t-1} + \\Phi_1 (Y_{t-1} - Y_{t-2} )- \\Theta_1 e_{t-1} - \\Theta_1 e_{t-1}$ Damped-trend linear Exponential smoothing|\n",
     "|ARIMA (0, 2, 1) OR (0,2,2) |0 2 1 |$\\hat Y_t = 2 Y_{t-1} - Y_{t-2} - \\Theta_1 e_{t-1} - \\Theta_2 e_{t-2}$|Linear exponential smoothing|\n",

nbs/docs/models/AutoARIMA.ipynb

@@ -105,9 +105,9 @@
     "|ARIMA (2, 0, 0)|2   0   0 |$\\hat Y_t = \\Phi_0 + \\Phi_1 Y_{t-1} + \\Phi_2 Y_{t-2} + \\epsilon$| AR(2): Second-order regression model|\n",
     "|ARIMA (1, 1, 0)|1   1   0 |$\\hat Y_t = \\mu + Y_{t-1} + \\Phi_1 (Y_{t-1}- Y_{t-2})$ | Differenced first-order autoregressive model|\n",
     "|ARIMA (0, 1, 1)|0   1   1 |$\\hat Y_t = Y_{t-1} - \\Phi_1 e^{t-1}$|Simple exponential smoothing|\n",
-    "|ARIMA (0, 0, 1)|0   0   1 |$\\hat Y_t = \\mu_0+ \\epsilon_t – \\omega_1 \\epsilon_{t-1}$|MA(1): First-order regression model|\n",
-    "ARIMA (0, 0, 2) |0   0   2 |$\\hat Y_t = \\mu_0+ \\epsilon_t – \\omega_1 \\epsilon_{t-1} – \\omega_2 \\epsilon_{t-2}$|MA(2): Second-order regression model|\n",
-    "|ARIMA (1, 0, 1)|1   0   1 |$\\hat Y_t = \\Phi_0 + \\Phi_1 Y_{t-1}+ \\epsilon_t – \\omega_1 \\epsilon_{t-1}$|ARMA model|\n",
+    "|ARIMA (0, 0, 1)|0   0   1 |$\\hat Y_t = \\mu_0+ \\epsilon_t - \\omega_1 \\epsilon_{t-1}$|MA(1): First-order regression model|\n",
+    "ARIMA (0, 0, 2) |0   0   2 |$\\hat Y_t = \\mu_0+ \\epsilon_t - \\omega_1 \\epsilon_{t-1} - \\omega_2 \\epsilon_{t-2}$|MA(2): Second-order regression model|\n",
+    "|ARIMA (1, 0, 1)|1   0   1 |$\\hat Y_t = \\Phi_0 + \\Phi_1 Y_{t-1}+ \\epsilon_t - \\omega_1 \\epsilon_{t-1}$|ARMA model|\n",
     "|ARIMA (1, 1, 1)|1   1   1 |$\\Delta Y_t = \\Phi_1 Y_{t-1} + \\epsilon_t - \\omega_1 \\epsilon_{t-1}$ |ARIMA model|\n",
     "|ARIMA (1, 1, 2)|1   1   2 |$\\hat Y_t = Y_{t-1} + \\Phi_1 (Y_{t-1} - Y_{t-2} )- \\Theta_1 e_{t-1} - \\Theta_1 e_{t-1}$ | Damped-trend linear Exponential smoothing|\n",
     "|ARIMA (0, 2, 1) OR (0,2,2) |0 2 1 |$\\hat Y_t = 2 Y_{t-1} - Y_{t-2} - \\Theta_1 e_{t-1} - \\Theta_2 e_{t-2}$|Linear exponential smoothing|\n",