I am trying to setup test environment for a react project written in typescript. I have done it before for react but with ES6.
Following is how the relevant parts of my package.json look like -
 "scripts": {
    "test": "NODE_ENV=test jest",
    "build": "webpack --mode production",
    "start": "webpack-dev-server --open --mode development"
  },
  "jest": {
    "moduleNameMapper": {
      "\\.(css)$": "jest-css-modules"
    },
    "transform": {
      "\\.(ts|tsx)$": "ts-jest"
    },
    "testRegex": ".test.(ts|tsx?)$",
    "moduleFileExensions": ["ts", "tsx", "js", "jsx", "json", "node"]
  },
I would like to mention that I am counting on ts transpiler and not configured babel at all.
Following are the dependencies I have included -
"dependencies": {
    "@types/jest": "^23.1.4",
    "@types/react": "^16.4.5",
    "@types/react-dom": "^16.0.6",
    "@types/react-redux": "^6.0.3",
    "@types/redux": "^3.6.0",
    "deep-freeze": "0.0.1",
    "jest": "^23.2.0",
    "jest-css-modules": "^1.1.0",
    "react-redux": "^5.0.7",
    "redux": "^4.0.0",
    "ts-jest": "^23.0.0",
    "webpack": "^4.12.2"
  },
  "devDependencies": {
    "awesome-typescript-loader": "^5.2.0",
    "css-loader": "^0.28.11",
    "html-loader": "^0.5.5",
    "html-webpack-plugin": "^3.2.0",
    "prop-types": "^15.6.2",
    "react": "^16.4.1",
    "react-dom": "^16.4.1",
    "style-loader": "^0.21.0",
    "typescript": "^2.9.2",
    "typings-for-css-modules-loader": "^1.7.0",
    "url-loader": "^1.0.1",
    "webpack-cli": "^3.0.8",
    "webpack-dev-server": "^3.1.4"
  }
I am unable to understand following error when I run npm run test as I have already installed and specified ts-jest in transform -
Validation Error:
  Module ts-jest in the transform option was not found.
         <rootDir> is: /Users/rahul/ot/Lab/webpack-playlist
Following is how tsconfig.json looks like
{
  "compilerOptions": {
    "jsx": "react",
    "noImplicitAny": true,
    "sourceMap": true,
    "target": "es5"
  }
}
Not sure how I can I resolve the error above.
I also tried configuring ts-jest in tsconfig.json like below but it still gave same error -
// tsconfig.json
    "jest": {
        "globals": {
          "ts-jest": {
            "tsConfigFile": "tsconfig.jest.json"
          }
        }
      }
// tsconfig.jest.json
{
  "extends": "./tsconfig",
  "compilerOptions": {
    "module": "commonjs"
  }
}
                Jest supports TypeScript, via Babel. First, make sure you followed the instructions on using Babel above. Next, install the @babel/preset-typescript : npm.
When you run jest with a jest. config. ts file it will use ts-node to compile that file, then it will pass it to ts-jest which will compile your tests, then it will pass those .
Looking through this, there isnt much that is different from my setup here:
https://github.com/jasonraimondi/typescript-react-starter
There is minor differences in your jest config. Below is the contents of my jest.config.js file. 
module.exports = {
  "transform": {
    "^.+\\.tsx?$": "ts-jest"
  },
  "testRegex": "(/__tests__/.*|(\\.|/)(test|spec))\\.(jsx?|tsx?)$",
  "moduleFileExtensions": [
    "ts",
    "tsx",
    "js",
    "jsx",
    "json",
    "node"
  ],
  "setupFiles": [
    "<rootDir>/test/setupTests.ts"
  ]
};
I have a post about testing a Typescript React App using ts-jest without "create-react-app" that hopefully may be of some help to.
Here is an example setup for using Jest in a TypeScript React project. I created it myself and I hope it helps you:
jest.config.js
const {defaults} = require('jest-config');
module.exports = {
  bail: true,
  moduleFileExtensions: [...defaults.moduleFileExtensions, 'ts', 'tsx'],
  roots: ['src'],
  testMatch: ['<rootDir>/src/**/?(*.)test.{ts,tsx}'],
  transform: {
    '^.+\\.tsx?$': 'ts-jest',
  },
  verbose: true,
};
package.json
{
  "dependencies": {
    "@types/react": "16.7.18",
    "@types/react-dom": "16.0.11",
    "react": "16.7.0",
    "react-dom": "16.7.0"
  },
  "devDependencies": {
    "jest": "23.6.0",
    "ts-jest": "23.10.5"
  },
  "scripts": {
    "test": "jest"
  },
  "version": "0.0.0"
}
src/Welcome.tsx
import * as React from 'react';
interface WelcomeProps {
  name: string,
}
const Welcome: React.FunctionComponent<WelcomeProps> = (props: WelcomeProps) => {
  return <h1>Hello, {props.name}</h1>;
};
export default Welcome;
src/Welcome.test.tsx
import React from 'react';
import ReactDOM from 'react-dom';
import Welcome from './Welcome';
it('renders without crashing', () => {
  const div = document.createElement('div');
  ReactDOM.render(<Welcome name="John Doe"/>, div);
  ReactDOM.unmountComponentAtNode(div);
});
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