Constructor for the quasi_hyperbolic-class — quasi

Defines an instance of the quasi_hyperbolic-class, that is the class defining the quasi-hyperbolic discounting model. For the mathematical equations and how to specify them in this constructor, users can look at the details. For more information on the model itself, see the vignette on the model definitions.

Usage

quasi_hyperbolic(
  d = NA,
  k = NA,
  parameters = NULL,
  covariance = NULL,
  cholesky = FALSE
)

Arguments

d: Integer denoting the number of dimensions of the model. Defaults to NA, in which case this dimensionality is inferred from the parameters. Note that if d and k are both defined while parameters is left unspecified, then the model will automatically create an empty model of the specified dimensionality.
k: Integer denoting the number of independent variables. Defaults to NA, in which case this dimensionality is inferred from the parameters.
parameters: List containing the parameters relevant to a particular model instance. Defaults to NULL, creating an empty model with $d = 1$ and $k = 1$ (unless d and k are specified).
covariance: Numeric matrix denoting the residual covariance of the model. Default to NULL, creating a matrix of 0s with the dimensionality implied by d.
cholesky: Logical denoting whether covariance is a lower-triangular decomposition matrix instead of an actual covariance matrix. Defaults to FALSE.

Value

Instance of quasi_hyperbolic-class

Details

The quasi-hyperbolic discounting model assumes that the effect of stimuli on affect fades away at a quasi-hyperbolic rate, that is first decreasing by a certain proportion after the first iteration and then decreasing at an exponential rate. The model can be defined as:

$$\boldsymbol{y}_t = \boldsymbol{\alpha} + \sum_{j = 0}^t N^j K^i(j) B \boldsymbol{x}_{t - j} + \boldsymbol{\epsilon}_t$$

where $i$ is an indicator function that is $0$ when $j = 0$ and $1$ otherwise, $\boldsymbol{\alpha}$ is a $d$-dimensional vector representing the mean of the process $\boldsymbol{y}$, $N$ and $K$ are $d \times d$ matrices containing the forgetting factors, determining how long the effect of the independent variables $\boldsymbol{x}$ on the process $\boldsymbol{y}$ lingers on and therefore determining the dynamics of the system, $B$ is a $d \times k$matrix containing the slopes of the independent variables $\boldsymbol{x}$, and $\boldsymbol{\epsilon}$ represents the residuals of the system. Within this package, we assume that:

$$\boldsymbol{\epsilon} \overset{iid}{\sim} N(\boldsymbol{0}, \Sigma)$$

where $\Sigma$ is a $d \times d$ matrix representing the residual covariance matrix.

The quasi-hyperbolic discounting model can also be written as an instance of the $VARMAX(1, 1)$, so that:

$$\boldsymbol{y}_t = (I_d - N) \boldsymbol{\alpha} + B \boldsymbol{x}_t N (K - I_d) B \boldsymbol{x}_{t - 1} + N \boldsymbol{y}_{t - 1} - N \boldsymbol{\epsilon}_{t - 1} + \boldsymbol{\epsilon}_t$$

where $I_d$ is a $d \times d$ identity matrix.

Interestingly, in multiple dimensions ($d > 1$), there is an alternative definition of the quasi-hyperbolic discounting model, specifically one where the matrices $N$ and $K$ are flipped, so that:

$$\boldsymbol{y}_t = \boldsymbol{\alpha} + \sum_{j = 0}^t \K^i(j) N^j B \boldsymbol{x}_{t - j} + \boldsymbol{\epsilon}_t$$

Its $VARMAX(1, 1)$ representation then becomes:

<TO DO>

For simplicity, we only use the first definition of the quasi-hyperbolic model within this package.

To define the model, one should minimally define the parameters of the model through parameters and covariance. In covariance, one either provides the $d \times d$ residual covariance matrix $\Sigma$ (cholesky = FALSE) or a lower-triangular $d \times d$ matrix $G$ that represents part of the decomposition of this matrix (cholesky = TRUE), so that we can retrieve the covariance matrix as:

$$\Sigma = G G^T$$

The latter option ensures that the covariance matrix $\Sigma$ defined within the model is positive-definite, but may require some additional thinking on the side of the user.

In parameters, one should provide a named list with instances "alpha", "beta", "nu", and "kappa", each defining the respective parameters $\boldsymbol{\alpha}$, $B$, $N$ and $K$ of the model. Note that the dimensionalities of these parameters should match up and are actively checked when constructing this class.

There are some important restrictions to take into account when defining the model:

The eigenvalues $\lambda_i$ of $N$ and $K$ should all lie between 0 and 1, or $\lambda_i \in [0, 1)$, a restriction that imposes exponential decay rather than sawtooth or more complex dynamics
The covariance matrix $\Sigma$ should be positive definite, as explained above

Examples

quasi_hyperbolic(
  d = 2,
  k = 2,
  parameters = list(
    "alpha" = numeric(2),
    "beta" = diag(2) * 2,
    "nu" = diag(2) * 0.5,
    "kappa" = diag(2) * 0.75
  ),
  covariance = diag(2)
)
#> Model of class "quasi_hyperbolic":
#> 
#> Dimension: 2
#> Number of predictors: 2
#> Number of parameters: 17
#> 
#> Parameters:
#>   alpha: |  0.00  |
#>          |  0.00  |
#> 
#>   beta: | 2.00  0.00 |
#>         | 0.00  2.00 |
#> 
#>   nu: | 0.50  0.00 |
#>       | 0.00  0.50 |
#> 
#>   kappa: | 0.75  0.00 |
#>          | 0.00  0.75 |
#> 
#> 
#> Covariance: | 1.00  0.00 |
#>             | 0.00  1.00 |

quasi_hyperbolic(
  d = 2,
  k = 2
)
#> Model of class "quasi_hyperbolic":
#> 
#> Dimension: 2
#> Number of predictors: 2
#> Number of parameters: 17
#> 
#> Parameters:
#>   alpha: |  0.00  |
#>          |  0.00  |
#> 
#>   beta: | 0.00  0.00 |
#>         | 0.00  0.00 |
#> 
#>   nu: | 0.00  0.00 |
#>       | 0.00  0.00 |
#> 
#>   kappa: | 0.00  0.00 |
#>          | 0.00  0.00 |
#> 
#> 
#> Covariance: | 0.00  0.00 |
#>             | 0.00  0.00 |

Constructor for the `quasi_hyperbolic-class`

Usage

Arguments

Value

Details

See also

Examples