Point Estimation Methods

Point Estimation

• An estimator is a function of the sample of data that provides an estimation of a parameter.
• Properties of good estimators: Unbiasedness, Small Variance, Consistency, and Efficiency.
• An estimator is unbiased if its expected value equals the true parameter value.
• Bias is the difference between the expected value of the estimator and the true parameter value: $Bias(\hat{\Theta}) = E[\hat{\Theta}] - \theta$.
• Mean Squared Error (MSE) measures the average squared difference between the estimator and the true parameter: $MSE(\hat{\Theta}) = E[(\hat{\Theta} - \theta)^2] = Var(\hat{\Theta}) + Bias(\hat{\Theta})^2$.
• Estimation methods include the Method of Moments and Maximum Likelihood Estimation (MLE).

Method of Moments

• Method of Moments equates sample moments with population moments to estimate parameters.

Maximum Likelihood Estimation (MLE)

• Maximum Likelihood Estimation (MLE) involves choosing parameters to maximize the likelihood function.
• Likelihood Function: $L(\theta; x_1,..., x_n) = \prod_{i=1}^{n} f(x_i; \theta)$ for independent and identically distributed (i.i.d) samples.
• Log-Likelihood Function: $\ell(\theta; x_1,..., x_n) = \sum_{i=1}^{n} log(f(x_i; \theta))$.
• Steps to finding MLE:
  • Write the likelihood function.
  • Take its logarithm.
  • Differentiate the log-likelihood function with respect to the parameter, setting it to zero.
  • Solve for the parameter to obtain the MLE.
  • Verify that the solution is a maximum.
• MLE is a function of the sample.
• Invariance Property: If $\hat{\theta}$ is the MLE of $\theta$, then $g(\hat{\theta})$ is the MLE of $g(\theta)$ for any function $g$.
• Under certain regularity conditions, MLEs are Consistent, Asymptotically Normal, and Efficient.

Cramer-Rao Lower Bound (CRLB)

• The Cramer-Rao Lower Bound (CRLB) provides a lower bound on the variance of unbiased estimators: $Var(\hat{\Theta}) \geq \frac{1}{nI(\theta)}$.
• $I(\theta)$ is the Fisher Information, which can be calculated as $E[(\frac{\partial}{\partial \theta} log f(X; \theta))^2]$ or $-E[\frac{\partial^2}{\partial \theta^2} log f(X; \theta)]$.

Matrizes

• A matrix $A_{m \times n}$ is an array of $m \cdot n$ elements in $m$ rows and $n$ columns.

Types of Matrices

• Row Matrix: A matrix with only one row ($1 \times n$).
• Column Matrix: A matrix with only one column ($m \times 1$).
• Square Matrix: Where the number of rows equals to the number of columns ($n \times n$).
  • The main diagonal is comprised of elements $a_{ii}$.
  • The secondary diagonal is comprised of elements $a_{ij}$ where $i + j = n + 1$.
• Null Matrix: All elements are zero.
• Identity Matrix ($I_n$): Square matrix with $a_{ij} = 1$ if $i = j$ and $a_{ij} = 0$ if $i \neq j$.
• Transpose Matrix ($A^T$): Rows and columns are swapped; if $A = [a_{ij}]$, then $A^T = [a_{ji}]$.
• Symmetric Matrix: $A = A^T$
• Antisymmetric Matrix: $A = -A^T$

Operations with Matrices

• Addition and Subtraction: Can only occur between matrices of the same order and results in an elementwise addition or subtraction of the corresponding elements.
• Scalar Multiplication: Each element of the matrix is multiplied by a scalar.
• Matrix Multiplication: $A_{m \times n} \cdot B_{n \times p} = C_{m \times p}$, where $c_{ij}$ is derived from the sum of the products of the elements in the $i$-th row of $A$ and the $j$-th column of $B$.

Determinants

• A function that associates a real number to a square matrix.

Calculation

• Order 2: $det(A) = a_{11}a_{22} - a_{12}a_{21}$
• Order 3: The Rule of Sarrus can be used.

Properties

• If a row (or column) is zero, the determinant is zero.
• Swapping two rows (or columns) inverts the sign of the determinant.
• Multiplying a row (or column) by a scalar multiplies the determinant by the same scalar.
• $det(A^T) = det(A)$
• $det(A \cdot B) = det(A) \cdot det(B)$

Inverse Matrix

• $A^{-1}$ is the matrix that, when multiplied by $A$, results in the identity matrix ($A \cdot A^{-1} = A^{-1} \cdot A = I$).

Calculation

• $A^{-1} = \frac{1}{det(A)} \cdot Adj(A)$, where $Adj(A)$ is the adjugate of $A$.

Existence

• Exists if and only if $det(A) \neq 0$.

Systeme of Equations

• System of Equations Matrix Representation: $A \cdot X = B$, where:
  • $A$ is the coefficient Matrix
  • $X$ is the Matrix of Unknowns
  • $B$ is the Matrix of Independent Terms

Resolution

• Cramer's Rule: Uses determinant to find the values of the variables
• Row Echelon Form (Escalonamento): Transforms the system into a simple equivalent system
• Inverse: If $A$ is invertible, $X = A^{-1} \cdot B$.

Bernoulli's Principle

• Daniel Bernoulli discovered Bernoulli's principle, which stipulates that for an inviscid flow, fluid speed increase occurs simultaneously with a pressure decrease or potential energy decrease.

Fluid Flow

• Laminar Flow: Fluid flows in parallel layers with no disruption between the layers.
• Turbulent Flow: Fluid undergoes irregular fluctuations or mixing.

Bernoulli's Equation

• ( P ) represents the static pressure of the fluid.
• ( \rho ) is the fluid density.
• ( v ) is the fluid speed.
• ( g ) is the acceleration caused by gravity.
• ( h ) is the height.

Applications of Bernoulli's Principle

• Aircraft wings are shaped so that air flows faster over the top than underneath
  • The higher pressure below the wing produces lift.
• Race cars use Bernoulli's principle to create downforce and enhance traction.
• Chimneys are tall to exploit pressure drop from wind, aiding smoke draft.
• Atomizers use rapid air streams to reduce pressure, drawing liquid up the tube for dispersion.

Algorithmic Trading

• Computer programs use pre-defined instructions to execute trades.
• Algorithms consider various factors such as price, timing, and volume.

Benerfits

• Lower transaction costs.
• Faster order execution and enhanced market efficiency.
• Reduced human error and emotional influence.
• Strategy back-testing.

Order Execution Strategies Include

• Market Order: Immediately executed at the best available price.
• Limit Order: Executed only at a specific price or better, providing price control but no guaranteed execution.
• Stop Order: Triggered at a specific price level, used to limit losses or protect profits.
• VWAP: Averages price based on volume, dividing order into smaller portions.
  • Formula: $$VWAP = \frac{\sum_{i}(Price_{i} \times Volume_{i})}{\sum_{i}Volume_{i}}$$
• TWAP: Averages price over a specified time frame, reducing adverse selection risk.
  • Formula: $$TWAP = \frac{\sum_{i}Price_{i}}{n}$$
• Implementation Shortfall: Seeks to minimize the difference between the actual execution price and the decision price, taking into account market impact.
  • Formula: $$Implementation \ Shortfall = (End \ Portfolio \ Value - Benchmark \ Portfolio \ Value) - Transaction \ Costs $$

Mitigating Market Impact

• Minimize market impact by using order splitting, stealth algorithms, and careful timing.

High-Frequency Trading (HFT)

• Characterized by high speed, high turnover, and short-term positions, utilizing advanced technology.

Key Concerns

• Market manipulation and unfair advantages.
• Potential to contribute to flash crashes.

Key Regulation & Ethics

• Regulatory oversight from bodies such as the SEC and FINRA is in charge ofalgorithmic trading activities.
• Regulations promote fairness and transparency in trading practices.
• Essential to carefully manage algorithmic trading, understanding trading strategies and considering ethics.